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Theorem carden2a 8737
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 8738 are meant to replace carden 9318 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
carden2a (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴𝐵)

Proof of Theorem carden2a
StepHypRef Expression
1 df-ne 2797 . 2 ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2 ndmfv 6176 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3 eqeq1 2630 . . . . . . 7 ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅))
42, 3syl5ibr 236 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅))
54con1d 139 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card))
65imp 445 . . . 4 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7 cardid2 8724 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
86, 7syl 17 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9 cardid2 8724 . . . . . . 7 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
10 ndmfv 6176 . . . . . . 7 𝐴 ∈ dom card → (card‘𝐴) = ∅)
119, 10nsyl4 156 . . . . . 6 (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
1211ensymd 7952 . . . . 5 (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
13 breq2 4622 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
14 entr 7953 . . . . . . 7 ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
1514ex 450 . . . . . 6 (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵𝐴𝐵))
1613, 15syl6bi 243 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵𝐴𝐵)))
1712, 16syl5 34 . . . 4 ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → ((card‘𝐵) ≈ 𝐵𝐴𝐵)))
1817imp 445 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵𝐴𝐵))
198, 18mpd 15 . 2 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴𝐵)
201, 19sylan2b 492 1 (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  wne 2796  c0 3896   class class class wbr 4618  dom cdm 5079  cfv 5850  cen 7897  cardccrd 8706
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-er 7688  df-en 7901  df-card 8710
This theorem is referenced by:  card1  8739
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