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Theorem carden 9411
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 13176 and the finite-set-only hashen 13175.

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 3467). 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 8796). (Contributed by NM, 22-Oct-2003.)

Assertion
Ref Expression
carden ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem carden
StepHypRef Expression
1 numth3 9330 . . . . . 6 (𝐴𝐶𝐴 ∈ dom card)
21ad2antrr 762 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ∈ dom card)
3 cardid2 8817 . . . . 5 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4 ensym 8046 . . . . 5 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
52, 3, 43syl 18 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ≈ (card‘𝐴))
6 simpr 476 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) = (card‘𝐵))
7 numth3 9330 . . . . . . 7 (𝐵𝐷𝐵 ∈ dom card)
87ad2antlr 763 . . . . . 6 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐵 ∈ dom card)
98cardidd 9409 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐵) ≈ 𝐵)
106, 9eqbrtrd 4707 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) ≈ 𝐵)
11 entr 8049 . . . 4 ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐵) → 𝐴𝐵)
125, 10, 11syl2anc 694 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴𝐵)
1312ex 449 . 2 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) → 𝐴𝐵))
14 carden2b 8831 . 2 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
1513, 14impbid1 215 1 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030   class class class wbr 4685  dom cdm 5143  cfv 5926  cen 7994  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-ac2 9323
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-wrecs 7452  df-recs 7513  df-er 7787  df-en 7998  df-card 8803  df-ac 8977
This theorem is referenced by:  cardeq0  9412  ficard  9425
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