Theorem carden 10541

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 14304 and the finite-set-only hashen 14303.

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 3768). 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 9885). (Contributed by NM, 22-Oct-2003.)

Assertion

Ref	Expression
carden	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Proof of Theorem carden

Step	Hyp	Ref	Expression
1		numth3 10460	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ dom card)
2	1	ad2antrr 723	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ∈ dom card)
3		cardid2 9943	. . . . 5 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4		ensym 8994	. . . . 5 ⊢ ((card‘𝐴) ≈ 𝐴 → 𝐴 ≈ (card‘𝐴))
5	2, 3, 4	3syl 18	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ≈ (card‘𝐴))
6		simpr 484	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) = (card‘𝐵))
7		numth3 10460	. . . . . . 7 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ dom card)
8	7	ad2antlr 724	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐵 ∈ dom card)
9	8	cardidd 10539	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐵) ≈ 𝐵)
10	6, 9	eqbrtrd 5160	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) ≈ 𝐵)
11		entr 8997	. . . 4 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐵) → 𝐴 ≈ 𝐵)
12	5, 10, 11	syl2anc 583	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ≈ 𝐵)
13	12	ex 412	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) → 𝐴 ≈ 𝐵))
14		carden2b 9957	. 2 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
15	13, 14	impbid1 224	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   class class class wbr 5138  dom cdm 5666  ‘cfv 6533   ≈ cen 8931  cardccrd 9925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-ac2 10453

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-er 8698  df-en 8935  df-card 9929  df-ac 10106

This theorem is referenced by:  cardeq0  10542  ficard  10555