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Theorem carden2a 9907
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. This assertion and carden2b 9908 are meant to replace carden 10492 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 2941 . 2 ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2 ndmfv 6878 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3 eqeq1 2737 . . . . . . 7 ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅))
42, 3imbitrrid 245 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅))
54con1d 145 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card))
65imp 408 . . . 4 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7 cardid2 9894 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
86, 7syl 17 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9 breq2 5110 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
10 entr 8949 . . . . . 6 ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
1110ex 414 . . . . 5 (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵𝐴𝐵))
129, 11syl6bi 253 . . . 4 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵𝐴𝐵)))
13 cardid2 9894 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
14 ndmfv 6878 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1513, 14nsyl4 158 . . . . 5 (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
1615ensymd 8948 . . . 4 (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
1712, 16impel 507 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → ((card‘𝐵) ≈ 𝐵𝐴𝐵))
188, 17mpd 15 . 2 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐴𝐵)
191, 18sylan2b 595 1 (((card‘𝐴) = (card‘𝐵) ∧ (card‘𝐴) ≠ ∅) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  c0 4283   class class class wbr 5106  dom cdm 5634  cfv 6497  cen 8883  cardccrd 9876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-er 8651  df-en 8887  df-card 9880
This theorem is referenced by:  card1  9909
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