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Type | Label | Description |
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Statement | ||
Theorem | xpcomen 6301 | Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 5-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | xpcomeng 6302 | Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.) |
Theorem | xpsnen2g 6303 | A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Theorem | xpassen 6304 | Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | xpdom2 6305 | Dominance law for Cartesian product. Proposition 10.33(2) of [TakeutiZaring] p. 92. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | xpdom2g 6306 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | xpdom1g 6307 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 25-Mar-2006.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | xpdom3m 6308* | A set is dominated by its Cartesian product with an inhabited set. Exercise 6 of [Suppes] p. 98. (Contributed by Jim Kingdon, 15-Apr-2020.) |
Theorem | xpdom1 6309 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.) |
Theorem | fopwdom 6310 | Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Theorem | enen1 6311 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
Theorem | enen2 6312 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
Theorem | domen1 6313 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Theorem | domen2 6314 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Theorem | phplem1 6315 | Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.) |
Theorem | phplem2 6316 | Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Theorem | phplem3 6317 | Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6319. (Contributed by NM, 26-May-1998.) |
Theorem | phplem4 6318 | Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Theorem | phplem3g 6319 | A natural number is equinumerous to its successor minus any element of the successor. Version of phplem3 6317 with unnecessary hypotheses removed. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Theorem | nneneq 6320 | Two equinumerous natural numbers are equal. Proposition 10.20 of [TakeutiZaring] p. 90 and its converse. Also compare Corollary 6E of [Enderton] p. 136. (Contributed by NM, 28-May-1998.) |
Theorem | php5 6321 | A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.) |
Theorem | snnen2og 6322 | A singleton is never equinumerous with the ordinal number 2. If is a proper class, see snnen2oprc 6323. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Theorem | snnen2oprc 6323 | A singleton is never equinumerous with the ordinal number 2. If is a set, see snnen2og 6322. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Theorem | phplem4dom 6324 | Dominance of successors implies dominance of the original natural numbers. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Theorem | php5dom 6325 | A natural number does not dominate its successor. (Contributed by Jim Kingdon, 1-Sep-2021.) |
Theorem | nndomo 6326 | Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.) |
Theorem | phpm 6327* | Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. By "proper subset" here we mean that there is an element which is in the natural number and not in the subset, or in symbols (which is stronger than not being equal in the absence of excluded middle). Theorem (Pigeonhole Principle) of [Enderton] p. 134. The theorem is so-called because you can't put n + 1 pigeons into n holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 6315 through phplem4 6318, nneneq 6320, and this final piece of the proof. (Contributed by NM, 29-May-1998.) |
Theorem | phpelm 6328 | Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.) |
Theorem | phplem4on 6329 | Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.) |
Theorem | fidceq 6330 | Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that is finite would require showing it is equinumerous to or to but to show that you'd need to know or , respectively. (Contributed by Jim Kingdon, 5-Sep-2021.) |
DECID | ||
Theorem | fidifsnen 6331 | All decrements of a finite set are equinumerous. (Contributed by Jim Kingdon, 9-Sep-2021.) |
Theorem | fidifsnid 6332 | If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3510 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.) |
Theorem | nnfi 6333 | Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Theorem | enfi 6334 | Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.) |
Theorem | enfii 6335 | A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) |
Theorem | ssfiexmid 6336* | If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.) |
Theorem | dif1en 6337 | If a set is equinumerous to the successor of a natural number , then with an element removed is equinumerous to . (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) |
Theorem | fiunsnnn 6338 | Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.) |
Theorem | php5fin 6339 | A finite set is not equinumerous to a set which adds one element. (Contributed by Jim Kingdon, 13-Sep-2021.) |
Theorem | fisbth 6340 | Schroeder-Bernstein Theorem for finite sets. (Contributed by Jim Kingdon, 12-Sep-2021.) |
Theorem | 0fin 6341 | The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
Theorem | fin0 6342* | A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.) |
Theorem | fin0or 6343* | A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.) |
Theorem | diffitest 6344* | If subtracting any set from a finite set gives a finite set, any proposition of the form is decidable. This is not a proof of full excluded middle, but it is close enough to show we won't be able to prove . (Contributed by Jim Kingdon, 8-Sep-2021.) |
Theorem | findcard 6345* | Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Theorem | findcard2 6346* | Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) |
Theorem | findcard2s 6347* | Variation of findcard2 6346 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.) |
Theorem | findcard2d 6348* | Deduction version of findcard2 6346. If you also need (which doesn't come for free due to ssfiexmid 6336), use findcard2sd 6349 instead. (Contributed by SO, 16-Jul-2018.) |
Theorem | findcard2sd 6349* | Deduction form of finite set induction . (Contributed by Jim Kingdon, 14-Sep-2021.) |
Theorem | diffisn 6350 | Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.) |
Theorem | diffifi 6351 | Subtracting one finite set from another produces a finite set. (Contributed by Jim Kingdon, 8-Sep-2021.) |
Theorem | ac6sfi 6352* | Existence of a choice function for finite sets. (Contributed by Jeff Hankins, 26-Jun-2009.) (Proof shortened by Mario Carneiro, 29-Jan-2014.) |
Theorem | fientri3 6353 | Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.) |
Theorem | nnwetri 6354* | A natural number is well-ordered by . More specifically, this order both satisfies and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.) |
Theorem | onunsnss 6355 | Adding a singleton to create an ordinal. (Contributed by Jim Kingdon, 20-Oct-2021.) |
Theorem | snon0 6356 | An ordinal which is a singleton is . (Contributed by Jim Kingdon, 19-Oct-2021.) |
Theorem | ordiso2 6357 | Generalize ordiso 6358 to proper classes. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Theorem | ordiso 6358* | Order-isomorphic ordinal numbers are equal. (Contributed by Jeff Hankins, 16-Oct-2009.) (Proof shortened by Mario Carneiro, 24-Jun-2015.) |
Syntax | ccrd 6359 | Extend class definition to include the cardinal size function. |
Definition | df-card 6360* | Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function. (Contributed by NM, 21-Oct-2003.) |
Theorem | cardcl 6361* | The cardinality of a well-orderable set is an ordinal. (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | isnumi 6362 | A set equinumerous to an ordinal is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Theorem | finnum 6363 | Every finite set is numerable. (Contributed by Mario Carneiro, 4-Feb-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | onenon 6364 | Every ordinal number is numerable. (Contributed by Mario Carneiro, 29-Apr-2015.) |
Theorem | cardval3ex 6365* | The value of . (Contributed by Jim Kingdon, 30-Aug-2021.) |
Theorem | oncardval 6366* | The value of the cardinal number function with an ordinal number as its argument. (Contributed by NM, 24-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Theorem | cardonle 6367 | The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
Theorem | card0 6368 | The cardinality of the empty set is the empty set. (Contributed by NM, 25-Oct-2003.) |
Theorem | carden2bex 6369* | If two numerable sets are equinumerous, then they have equal cardinalities. (Contributed by Jim Kingdon, 30-Aug-2021.) |
This section derives the basics of real and complex numbers. To construct the real numbers constructively, we follow two main sources. The first is Metamath Proof Explorer, which has the advantage of being already formalized in metamath. Its disadvantage, for our purposes, is that it assumes the law of the excluded middle throughout. Since we have already developed natural numbers ( for example, nna0 6053 and similar theorems ), going from there to positive integers (df-ni 6402) and then positive rational numbers (df-nqqs 6446) does not involve a major change in approach compared with the Metamath Proof Explorer. It is when we proceed to Dedekind cuts that we bring in more material from Section 11.2 of [HoTT], which focuses on the aspects of Dedekind cuts which are different without excluded middle. With excluded middle, it is natural to define the cut as the lower set only (as Metamath Proof Explorer does), but we define the cut as a pair of both the lower and upper sets, as [HoTT] does. There are also differences in how we handle order and replacing "not equal to zero" with "apart from zero". | ||
Syntax | cnpi 6370 |
The set of positive integers, which is the set of natural numbers
with 0 removed.
Note: This is the start of the Dedekind-cut construction of real and _complex numbers. |
Syntax | cpli 6371 | Positive integer addition. |
Syntax | cmi 6372 | Positive integer multiplication. |
Syntax | clti 6373 | Positive integer ordering relation. |
Syntax | cplpq 6374 | Positive pre-fraction addition. |
Syntax | cmpq 6375 | Positive pre-fraction multiplication. |
Syntax | cltpq 6376 | Positive pre-fraction ordering relation. |
Syntax | ceq 6377 | Equivalence class used to construct positive fractions. |
Syntax | cnq 6378 | Set of positive fractions. |
Syntax | c1q 6379 | The positive fraction constant 1. |
Syntax | cplq 6380 | Positive fraction addition. |
Syntax | cmq 6381 | Positive fraction multiplication. |
Syntax | crq 6382 | Positive fraction reciprocal operation. |
Syntax | cltq 6383 | Positive fraction ordering relation. |
Syntax | ceq0 6384 | Equivalence class used to construct non-negative fractions. |
~_{Q0} | ||
Syntax | cnq0 6385 | Set of non-negative fractions. |
Q_{0} | ||
Syntax | c0q0 6386 | The non-negative fraction constant 0. |
0_{Q0} | ||
Syntax | cplq0 6387 | Non-negative fraction addition. |
+_{Q0} | ||
Syntax | cmq0 6388 | Non-negative fraction multiplication. |
·_{Q0} | ||
Syntax | cnp 6389 | Set of positive reals. |
Syntax | c1p 6390 | Positive real constant 1. |
Syntax | cpp 6391 | Positive real addition. |
Syntax | cmp 6392 | Positive real multiplication. |
Syntax | cltp 6393 | Positive real ordering relation. |
Syntax | cer 6394 | Equivalence class used to construct signed reals. |
Syntax | cnr 6395 | Set of signed reals. |
Syntax | c0r 6396 | The signed real constant 0. |
Syntax | c1r 6397 | The signed real constant 1. |
Syntax | cm1r 6398 | The signed real constant -1. |
Syntax | cplr 6399 | Signed real addition. |
Syntax | cmr 6400 | Signed real multiplication. |
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