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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | domtr 6201 | Transitivity of dominance relation. Theorem 17 of [Suppes] p. 94. (Contributed by NM, 4-Jun-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | entri 6202 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Theorem | entr2i 6203 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Theorem | entr3i 6204 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Theorem | entr4i 6205 | A chained equinumerosity inference. (Contributed by NM, 25-Sep-2004.) |
Theorem | endomtr 6206 | Transitivity of equinumerosity and dominance. (Contributed by NM, 7-Jun-1998.) |
Theorem | domentr 6207 | Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
Theorem | f1imaeng 6208 | A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by Mario Carneiro, 15-May-2015.) |
Theorem | f1imaen2g 6209 | A one-to-one function's image under a subset of its domain is equinumerous to the subset. (This version of f1imaen 6210 does not need ax-setind 4220.) (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Theorem | f1imaen 6210 | A one-to-one function's image under a subset of its domain is equinumerous to the subset. (Contributed by NM, 30-Sep-2004.) |
Theorem | en0 6211 | The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) |
Theorem | ensn1 6212 | A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) |
Theorem | ensn1g 6213 | A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
Theorem | enpr1g 6214 | has only one element. (Contributed by FL, 15-Feb-2010.) |
Theorem | en1 6215* | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) |
Theorem | en1bg 6216 | A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Theorem | reuen1 6217* | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Theorem | euen1 6218 | Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) |
Theorem | euen1b 6219* | Two ways to express " has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.) |
Theorem | en1uniel 6220 | A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Theorem | 2dom 6221* | A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
Theorem | fundmen 6222 | A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 28-Jul-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | fundmeng 6223 | A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.) |
Theorem | cnven 6224 | A relational set is equinumerous to its converse. (Contributed by Mario Carneiro, 28-Dec-2014.) |
Theorem | fndmeng 6225 | A function is equinumerate to its domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
Theorem | en2sn 6226 | Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) |
Theorem | snfig 6227 | A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Theorem | fiprc 6228 | The class of finite sets is a proper class. (Contributed by Jeff Hankins, 3-Oct-2008.) |
Theorem | unen 6229 | Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Theorem | enm 6230* | A set equinumerous to an inhabited set is inhabited. (Contributed by Jim Kingdon, 19-May-2020.) |
Theorem | xpsnen 6231 | A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 4-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Theorem | xpsneng 6232 | A set is equinumerous to its Cartesian product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254. (Contributed by NM, 22-Oct-2004.) |
Theorem | xp1en 6233 | One times a cardinal number. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Theorem | endisj 6234* | Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.) |
Theorem | xpcomf1o 6235* | The canonical bijection from to . (Contributed by Mario Carneiro, 23-Apr-2014.) |
Theorem | xpcomco 6236* | Composition with the bijection of xpcomf1o 6235 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.) |
Theorem | xpcomen 6237 | 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 6238 | Commutative law for equinumerosity of Cartesian product. Proposition 4.22(d) of [Mendelson] p. 254. (Contributed by NM, 27-Mar-2006.) |
Theorem | xpsnen2g 6239 | A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Theorem | xpassen 6240 | 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 6241 | 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 6242 | Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Theorem | xpdom1g 6243 | 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 6244* | 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 6245 | 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 6246 | Covering implies injection on power sets. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Theorem | enen1 6247 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
Theorem | enen2 6248 | Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.) |
Theorem | domen1 6249 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Theorem | domen2 6250 | Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.) |
Theorem | nnfi 6251 | Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Theorem | enfi 6252 | Equinumerous sets have the same finiteness. (Contributed by NM, 22-Aug-2008.) |
Theorem | enfii 6253 | A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) |
Theorem | ssfiexmid 6254* | 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 | 0fin 6255 | The empty set is finite. (Contributed by FL, 14-Jul-2008.) |
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 5992 and similar theorems ), going from there to positive integers (df-ni 6288) and then positive rational numbers (df-nqqs 6332) 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 6256 |
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 6257 | Positive integer addition. |
Syntax | cmi 6258 | Positive integer multiplication. |
Syntax | clti 6259 | Positive integer ordering relation. |
Syntax | cplpq 6260 | Positive pre-fraction addition. |
Syntax | cmpq 6261 | Positive pre-fraction multiplication. |
Syntax | cltpq 6262 | Positive pre-fraction ordering relation. |
Syntax | ceq 6263 | Equivalence class used to construct positive fractions. |
Syntax | cnq 6264 | Set of positive fractions. |
Syntax | c1q 6265 | The positive fraction constant 1. |
Syntax | cplq 6266 | Positive fraction addition. |
Syntax | cmq 6267 | Positive fraction multiplication. |
Syntax | crq 6268 | Positive fraction reciprocal operation. |
Syntax | cltq 6269 | Positive fraction ordering relation. |
Syntax | ceq0 6270 | Equivalence class used to construct non-negative fractions. |
~_{Q0} | ||
Syntax | cnq0 6271 | Set of non-negative fractions. |
Q_{0} | ||
Syntax | c0q0 6272 | The non-negative fraction constant 0. |
0_{Q0} | ||
Syntax | cplq0 6273 | Non-negative fraction addition. |
+_{Q0} | ||
Syntax | cmq0 6274 | Non-negative fraction multiplication. |
·_{Q0} | ||
Syntax | cnp 6275 | Set of positive reals. |
Syntax | c1p 6276 | Positive real constant 1. |
Syntax | cpp 6277 | Positive real addition. |
Syntax | cmp 6278 | Positive real multiplication. |
Syntax | cltp 6279 | Positive real ordering relation. |
Syntax | cer 6280 | Equivalence class used to construct signed reals. |
Syntax | cnr 6281 | Set of signed reals. |
Syntax | c0r 6282 | The signed real constant 0. |
Syntax | c1r 6283 | The signed real constant 1. |
Syntax | cm1r 6284 | The signed real constant -1. |
Syntax | cplr 6285 | Signed real addition. |
Syntax | cmr 6286 | Signed real multiplication. |
Syntax | cltr 6287 | Signed real ordering relation. |
Definition | df-ni 6288 | Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 15-Aug-1995.) |
Definition | df-pli 6289 | Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
Definition | df-mi 6290 | Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers and is intended to be used only by the construction. (Contributed by NM, 26-Aug-1995.) |
Definition | df-lti 6291 | Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 6-Feb-1996.) |
Theorem | elni 6292 | Membership in the class of positive integers. (Contributed by NM, 15-Aug-1995.) |
Theorem | pinn 6293 | A positive integer is a natural number. (Contributed by NM, 15-Aug-1995.) |
Theorem | pion 6294 | A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) |
Theorem | piord 6295 | A positive integer is ordinal. (Contributed by NM, 29-Jan-1996.) |
Theorem | niex 6296 | The class of positive integers is a set. (Contributed by NM, 15-Aug-1995.) |
Theorem | 0npi 6297 | The empty set is not a positive integer. (Contributed by NM, 26-Aug-1995.) |
Theorem | elni2 6298 | Membership in the class of positive integers. (Contributed by NM, 27-Nov-1995.) |
Theorem | 1pi 6299 | Ordinal 'one' is a positive integer. (Contributed by NM, 29-Oct-1995.) |
Theorem | addpiord 6300 | Positive integer addition in terms of ordinal addition. (Contributed by NM, 27-Aug-1995.) |
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