Theorem carden 8947

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 12421 and the finite-set-only hashen 12420.

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 3326). 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 8334). (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
carden

Proof of Theorem carden

Step	Hyp	Ref	Expression
1		numth3 8871	. . . . . 6
2	1	ad2antrr 725	. . . . 5
3		cardid2 8355	. . . . 5
4		ensym 7584	. . . . 5
5	2, 3, 4	3syl 20	. . . 4
6		simpr 461	. . . . 5
7		numth3 8871	. . . . . . 7
8	7	ad2antlr 726	. . . . . 6
9	8	cardidd 8945	. . . . 5
10	6, 9	eqbrtrd 4472	. . . 4
11		entr 7587	. . . 4
12	5, 10, 11	syl2anc 661	. . 3
13	12	ex 434	. 2
14		carden2b 8369	. 2
15	13, 14	impbid1 203	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   class class class wbr 4452  domcdm 5004  `cfv 5593  cen 7533  ccrd 8337

This theorem is referenced by:  cardeq0  8948  ficard  8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592  ax-ac2 8864

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-int 4287  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6257  df-recs 7061  df-er 7330  df-en 7537  df-card 8341  df-ac 8518