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Mirrors > Home > MPE Home > Th. List > carden | Structured version Visualization version Unicode version |
Description: Two sets are equinumerous
iff their cardinal numbers are equal. This
important theorem expresses the essential concept behind
"cardinality" or
"size." This theorem appears as Proposition 10.10 of [TakeutiZaring]
p. 85, Theorem 7P of [Enderton] p. 197,
and Theorem 9 of [Suppes] p. 242
(among others). The Axiom of Choice is required for its proof. Related
theorems are hasheni 13136 and the finite-set-only hashen 13135.
This theorem is also known as Hume's Principle. Gottlob Frege's two-volume Grundgesetze der Arithmetik used his Basic Law V to prove this theorem. Unfortunately Basic Law V caused Frege's system to be inconsistent because it was subject to Russell's paradox (see ru 3434). Later scholars have found that Frege primarily used Basic Law V to Hume's Principle. If Basic Law V is replaced by Hume's Principle in Frege's system, much of Frege's work is restored. Grundgesetze der Arithmetik, once Basic Law V is replaced, proves "Frege's theorem" (the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle). See https://plato.stanford.edu/entries/frege-theorem . We take a different approach, using first-order logic and ZFC, to prove the Peano axioms of arithmetic. The theory of cardinality can also be developed without AC by introducing "card" as a primitive notion and stating this theorem as an axiom, as is done with the axiom for cardinal numbers in [Suppes] p. 111. Finally, if we allow the Axiom of Regularity, we can avoid AC by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank (see karden 8758). (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
carden |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | numth3 9292 | . . . . . 6 | |
2 | 1 | ad2antrr 762 | . . . . 5 |
3 | cardid2 8779 | . . . . 5 | |
4 | ensym 8005 | . . . . 5 | |
5 | 2, 3, 4 | 3syl 18 | . . . 4 |
6 | simpr 477 | . . . . 5 | |
7 | numth3 9292 | . . . . . . 7 | |
8 | 7 | ad2antlr 763 | . . . . . 6 |
9 | 8 | cardidd 9371 | . . . . 5 |
10 | 6, 9 | eqbrtrd 4675 | . . . 4 |
11 | entr 8008 | . . . 4 | |
12 | 5, 10, 11 | syl2anc 693 | . . 3 |
13 | 12 | ex 450 | . 2 |
14 | carden2b 8793 | . 2 | |
15 | 13, 14 | impbid1 215 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 class class class wbr 4653 cdm 5114 cfv 5888 cen 7952 ccrd 8761 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-ac2 9285 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-wrecs 7407 df-recs 7468 df-er 7742 df-en 7956 df-card 8765 df-ac 8939 |
This theorem is referenced by: cardeq0 9374 ficard 9387 |
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