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Theorem carden2a 8752
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 8753 are meant to replace carden 9333 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 2791 . 2 ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2 ndmfv 6185 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3 eqeq1 2625 . . . . . . 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 8739 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
86, 7syl 17 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9 cardid2 8739 . . . . . . 7 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
10 ndmfv 6185 . . . . . . 7 𝐴 ∈ dom card → (card‘𝐴) = ∅)
119, 10nsyl4 156 . . . . . 6 (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
1211ensymd 7967 . . . . 5 (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
13 breq2 4627 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
14 entr 7968 . . . . . . 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 1987  wne 2790  c0 3897   class class class wbr 4623  dom cdm 5084  cfv 5857  cen 7912  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-card 8725
This theorem is referenced by:  card1  8754
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