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Theorem carden 9626
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 13340 and the finite-set-only hashen 13339.

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 3595). 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 8973). (Contributed by NM, 22-Oct-2003.)

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

Proof of Theorem carden
StepHypRef Expression
1 numth3 9545 . . . . . 6 (𝐴𝐶𝐴 ∈ dom card)
21ad2antrr 717 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ∈ dom card)
3 cardid2 9030 . . . . 5 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4 ensym 8209 . . . . 5 ((card‘𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴))
52, 3, 43syl 18 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ≈ (card‘𝐴))
6 simpr 477 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) = (card‘𝐵))
7 numth3 9545 . . . . . . 7 (𝐵𝐷𝐵 ∈ dom card)
87ad2antlr 718 . . . . . 6 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐵 ∈ dom card)
98cardidd 9624 . . . . 5 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐵) ≈ 𝐵)
106, 9eqbrtrd 4831 . . . 4 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) ≈ 𝐵)
11 entr 8212 . . . 4 ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐵) → 𝐴𝐵)
125, 10, 11syl2anc 579 . . 3 (((𝐴𝐶𝐵𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴𝐵)
1312ex 401 . 2 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) → 𝐴𝐵))
14 carden2b 9044 . 2 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
1513, 14impbid1 216 1 ((𝐴𝐶𝐵𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155   class class class wbr 4809  dom cdm 5277  cfv 6068  cen 8157  cardccrd 9012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-ac2 9538
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-wrecs 7610  df-recs 7672  df-er 7947  df-en 8161  df-card 9016  df-ac 9190
This theorem is referenced by:  cardeq0  9627  ficard  9640
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