Theorem carden 10473

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 14283 and the finite-set-only hashen 14282.

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 3740). 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 3740. 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 9819). (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 10392	. . . . . 6 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ dom card)
2	1	ad2antrr 727	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐴 ∈ dom card)
3		cardid2 9877	. . . . 5 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
4		ensym 8952	. . . . 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 10392	. . . . . . 7 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ dom card)
8	7	ad2antlr 728	. . . . . 6 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → 𝐵 ∈ dom card)
9	8	cardidd 10471	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐵) ≈ 𝐵)
10	6, 9	eqbrtrd 5122	. . . 4 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (card‘𝐴) = (card‘𝐵)) → (card‘𝐴) ≈ 𝐵)
11		entr 8955	. . . 4 ⊢ ((𝐴 ≈ (card‘𝐴) ∧ (card‘𝐴) ≈ 𝐵) → 𝐴 ≈ 𝐵)
12	5, 10, 11	syl2anc 585	. . 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 9891	. 2 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵))
15	13, 14	impbid1 225	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴 ≈ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  dom cdm 5632  ‘cfv 6500   ≈ cen 8892  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-er 8645  df-en 8896  df-card 9863  df-ac 10038

This theorem is referenced by:  cardeq0  10474  ficard  10487