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Theorem carden2 8773
 Description: Two numerable sets are equinumerous iff their cardinal numbers are equal. Unlike carden 9333, the Axiom of Choice is not required. (Contributed by Mario Carneiro, 22-Sep-2013.)
Assertion
Ref Expression
carden2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem carden2
StepHypRef Expression
1 carddom2 8763 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
2 carddom2 8763 . . . 4 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵𝐴))
32ancoms 469 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ 𝐵𝐴))
41, 3anbi12d 746 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (𝐴𝐵𝐵𝐴)))
5 eqss 3603 . . 3 ((card‘𝐴) = (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)))
65bicomi 214 . 2 (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐵) ⊆ (card‘𝐴)) ↔ (card‘𝐴) = (card‘𝐵))
7 sbthb 8041 . 2 ((𝐴𝐵𝐵𝐴) ↔ 𝐴𝐵)
84, 6, 73bitr3g 302 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ⊆ wss 3560   class class class wbr 4623  dom cdm 5084  ‘cfv 5857   ≈ cen 7912   ≼ cdom 7913  cardccrd 8721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-card 8725 This theorem is referenced by:  cardsdom2  8774  pm54.43lem  8785  sdom2en01  9084  fin23lem22  9109  fin1a2lem9  9190  pwfseqlem4  9444  hashen  13091
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