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Theorem carden 10589
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 14384 and the finite-set-only hashen 14383.

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 3789). 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 3789. 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 9933). (Contributed by NM, 22-Oct-2003.)

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

Proof of Theorem carden
StepHypRef Expression
1 numth3 10508 . . . . . 6 (𝐴𝐶𝐴 ∈ dom card)
21ad2antrr 726 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ∈ dom card)
3 cardid2 9991 . . . . 5 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4 ensym 9042 . . . . 5 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
52, 3, 43syl 18 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ≈ (card‘𝐴))
6 simpr 484 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) = (card‘𝐵))
7 numth3 10508 . . . . . . 7 (𝐵𝐷𝐵 ∈ dom card)
87ad2antlr 727 . . . . . 6 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐵 ∈ dom card)
98cardidd 10587 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐵) ≈ 𝐵)
106, 9eqbrtrd 5170 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) ≈ 𝐵)
11 entr 9045 . . . 4 ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐵) → 𝐴𝐵)
125, 10, 11syl2anc 584 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴𝐵)
1312ex 412 . 2 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) → 𝐴𝐵))
14 carden2b 10005 . 2 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
1513, 14impbid1 225 1 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106   class class class wbr 5148  dom cdm 5689  cfv 6563  cen 8981  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-er 8744  df-en 8985  df-card 9977  df-ac 10154
This theorem is referenced by:  cardeq0  10590  ficard  10603
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