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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxccsc 26901 Extend class notation to include the cosecant function, see df-csc 26904.

Syntaxccot 26902 Extend class notation to include the cotangent function, see df-cot 26905.

Definitiondf-sec 26903* Define the secant function. We define it this way for cmpt 3974, which requires the form . The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Definitiondf-csc 26904* Define the cosecant function. We define it this way for cmpt 3974, which requires the form . The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Definitiondf-cot 26905* Define the cotangent function. We define it this way for cmpt 3974, which requires the form . The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremsecval 26906 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcscval 26907 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcotval 26908 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremseccl 26909 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcsccl 26910 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremcotcl 26911 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremreseccl 26912 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecsccl 26913 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecotcl 26914 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)

Theoremrecsec 26915 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremreccsc 26916 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)

Theoremreccot 26917 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)

Theoremrectan 26918 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)

Theoremsec0 26919 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)

Theoremonetansqsecsq 26920 Prove the tangent squared secant squared identity A ) ^ 2 ) ) = ( ( sec . (Contributed by David A. Wheeler, 25-May-2015.)

Theoremcotsqcscsq 26921 Prove the tangent squared cosecant squared identity A ) ^ 2 ) ) = ( ( csc . (Contributed by David A. Wheeler, 27-May-2015.)

16.19.5  Identities for "if"

Utility theorems for "if".

Theoremifnmfalse 26922 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3477 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)

16.19.6  Not-member-of

TheoremAnelBC 26923 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using . (Contributed by David A. Wheeler, 10-May-2015.)

16.19.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 26927 and df-dp2 26926 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10004.

TODO: Fix non-existent label dfpval.

Syntaxcdp2 26924 Constant used for decimal fraction constructor. See df-dp2 26926.
_

Syntaxcdp 26925 Decimal point operator. See df-dp 26927.

Definitiondf-dp2 26926 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10004. (Contributed by David A. Wheeler, 15-May-2015.)
_

Definitiondf-dp 26927* Define the (decimal point) operator. For example, , and ;__ ;;;; ;;; Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is , not ; this should simplify some proofs. The LHS is , since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

_

Theoremdp2cl 26928 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
_

Theoremdpval 26929 Define the value of the decimal point operator. See df-dp 26927. (Contributed by David A. Wheeler, 15-May-2015.)
_

Theoremdpcl 26930 Prove that the closure of the decimal point is as we have defined it. See df-dp 26927. (Contributed by David A. Wheeler, 15-May-2015.)

Theoremdpfrac1 26931 Prove a simple equivalence involving the decimal point. See df-dp 26927 and dpcl 26930. (Contributed by David A. Wheeler, 15-May-2015.)
;

16.19.8  Signum (sgn or sign) function

Syntaxcsgn 26932 Extend class notation to include the Signum function.
sgn

Definitiondf-sgn 26933 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over (df-xr 8751) instead of so that it can accept and . Note that df-psgn 26581 defines the sign of a permutation, which is different. Value shown in sgnval 26934. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnval 26934 Value of Signum function. Pronounced "signum" . See df-sgn 26933. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgn0 26935 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnp 26936 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnrrp 26937 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
sgn

Theoremsgn1 26938 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theoremsgnpnf 26939 Proof that the signum of is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theoremsgnn 26940 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnmnf 26941 Proof that the signum of is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

16.19.9  Ceiling function

Syntaxccei 26942 Extend class notation to include the ceiling function.

Definitiondf-ceiling 26943 The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.

By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 10833 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.)

Theoremceilingval 26944 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)

Theoremceilingcl 26945 Closure of the ceiling function; the real work is in ceicl 10833. (Contributed by David A. Wheeler, 19-May-2015.)

16.19.10  Logarithm laws generalized to an arbitrary base

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

There are different ways this could be defined in Metamath. The approach used here is intentionally similar to existing 2-parameter Metamath functions. The way defined here supports two notations, logb and logb where is the base and is the other parameter. An alternative would be to support the notational form logb; that looks a little more like traditional notation, but is different than other 2-parameter functions. It's not obvious to me which is better, so here we try out one way as an experiment. Feedback and help welcome.

Syntaxclogb 26946 Extend class notation to include the logarithm generalized to an arbitrary base.
logb

Definitiondf-logb 26947* Define the logb operator. This is the logarithm generalized to an arbitrary base. It can be used as logb for "log base B of X". In the most common traditional notation, base B is a subscript of "log". You could also use logb, which looks like a less-common notation that some use where the base is a preceding superscript. Note: This definition doesn't prevent bases of 1 or 0; proofs may need to forbid them. (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

Theoremlogbnfxval 26948 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used as infix. Most Metamath statements select variables in order of their use, but to make the order clearer we use "B" for base and "X" for the other operand here. Proof is similar to modval 10853. (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

Theoremlogbval 26949 Define the value of the logb function, the logarithm generalized to an arbitrary base, when used in the form logb (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

Theoremreglogbcl 26950 General logarithm is a real number, given extended real numbers. Based on reglogcl 26141. (Contributed by David A. Wheeler, 21-Jan-2017.)
logb

16.19.11  Miscellaneous

Miscellaneous proofs.

Theorem2m1e1 26951 Prove that 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 9738. (Contributed by David A. Wheeler, 4-Jan-2017.)

Theorem5m4e1 26952 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)

Theorem2p2ne5 26953 Prove that . In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)

Theoremresolution 26954 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)

16.20  Mathbox for Alan Sare

16.20.1  Conventional Metamath proofs, some derived from VD proofs

Theoremiidn3 26955 idn3 27077 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee222 26956 e222 27098 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee3bir 26957 Right-biconditional form of e3 27202 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee13 26958 e13 27213 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee121 26959 e121 27118 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee122 26960 e122 27115 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee333 26961 e333 27198 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee323 26962 e323 27231 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3ornot23 26963 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 27313. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremorbi1r 26964 orbi1 689 with order of disjuncts reversed. Derived from orbi1rVD 27314. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorembitr3 26965 Closed nested implication form of bitr3i 244. Derived automatically from bitr3VD 27315. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3orbi123 26966 pm4.39 846 with a 3-conjunct antecedent. This proof is 3orbi123VD 27316 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsyl5imp 26967 Closed form of syl5 30. Derived automatically from syl5impVD 27329. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimpexp3a 26968 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: qed:1:

Theoremcom3rgbi 26969 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: 2:: 3:1,2: 4:: 5:: 6:4,5: qed:3,6:

Theoremimpexp3acom3r 26970 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: 2:: qed:1,2:

Theoremee1111 26971 Non-virtual deduction form of e1111 27137. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: h4:: h5:: 6:1,5: 7:6: 8:2,7: 9:8: 10:9: 11:3,10: 12:11: 13:12: 14:4,13: qed:14:

Theorempm2.43bgbi 26972 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theorempm2.43cbi 26973 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theoremee233 26974 Non-virtual deduction form of e233 27230. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: h4:: 5:1,4: 6:5: 7:2,6: 8:7: 9:8: 10:9: 11:10: 12:3,11: 13:12: 14:13: qed:14:

Theoremimbi12 26975 Implication form of imbi12i 318. imbi12 26975 is imbi12VD 27339 without virtual deductions and was automatically derived from imbi12VD 27339 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimbi13 26976 Join three logical equivalences to form equivalence of implications. imbi13 26976 is imbi13VD 27340 without virtual deductions and was automatically derived from imbi13VD 27340 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee33 26977 Non-virtual deduction form of e33 27199. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: 4:1,3: 5:4: 6:2,5: 7:6: 8:7: qed:8:

Theoremcon5 26978 Bi-conditional contraposition variation. This proof is con5VD 27366 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon5i 26979 Inference form of con5 26978. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexlimexi 26980 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsb5ALT 26981* Equivalence for substitution. Alternate proof of sb5 1993. This proof is sb5ALTVD 27379 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst01 26982 exinst01 27087 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst11 26983 exinst11 27088 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremvk15.4j 26984 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 26984 is vk15.4jVD 27380 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnotnot2ALT 26985 Converse of double negation. Alternate proof of notnot2 106. This proof is notnot2ALTVD 27381 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon3ALT 26986 Contraposition. Alternate proof of con3 128. This proof is con3ALTVD 27382 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremssralv2 26987* Quantification restricted to a subclass for two quantifiers. ssralv 3158 for two quantifiers. The proof of ssralv2 26987 was automatically generated by minimizing the automatically translated proof of ssralv2VD 27332. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbc3org 26988 sbcorg 2966 with a 3-disjuncts. This proof is sbc3orgVD 27317 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalrim3con13v 26989* Closed form of alrimi 1706 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 27318 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremra4sbc2 26990* ra4sbc 2999 with two quantifying variables. This proof is ra4sbc2VD 27321 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcoreleleq 26991* Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 27325. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtratrb 26992* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 27327. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3ax5 26993 ax-5 1533 for a 3 element left-nested implication. Derived automatically from 3ax5VD 27328. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremordelordALT 26994 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4307 using the Axiom of Regularity indirectly through dford2 7205. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that because this is inferred by the Axiom of Regularity. ordelordALT 26994 is ordelordALTVD 27333 without virtual deductions and was automatically derived from ordelordALTVD 27333 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcim2g 26995 Distribution of class substitution over a left-nested implication. Similar to sbcimg 2962. sbcim2g 26995 is sbcim2gVD 27341 without virtual deductions and was automatically derived from sbcim2gVD 27341 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcbi 26996 Implication form of sbcbiiOLD 2977. sbcbi 26996 is sbcbiVD 27342 without virtual deductions and was automatically derived from sbcbiVD 27342 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtrsbc 26997* Formula-building inference rule for class substitution, substituting a class variable for the set variable of the transitivity predicate. trsbc 26997 is trsbcVD 27343 without virtual deductions and was automatically derived from trsbcVD 27343 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremtruniALT 26998* The union of a class of transitive sets is transitive. Alternate proof of truni 4024. truniALT 26998 is truniALTVD 27344 without virtual deductions and was automatically derived from truniALTVD 27344 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcss 26999 Distribute proper substitution through a subclass relation. This theorem was automatically derived from sbcssVD 27349. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremonfrALTlem5 27000* Lemma for onfrALT 27007. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

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