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Theorem carden2a 9389
Description: If two sets have equal nonzero cardinalities, then they are equinumerous. (This assertion and carden2b 9390 are meant to replace carden 9967 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 3017 . 2 ((card‘𝐴) ≠ ∅ ↔ ¬ (card‘𝐴) = ∅)
2 ndmfv 6694 . . . . . . 7 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3 eqeq1 2825 . . . . . . 7 ((card‘𝐴) = (card‘𝐵) → ((card‘𝐴) = ∅ ↔ (card‘𝐵) = ∅))
42, 3syl5ibr 248 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (¬ 𝐵 ∈ dom card → (card‘𝐴) = ∅))
54con1d 147 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (¬ (card‘𝐴) = ∅ → 𝐵 ∈ dom card))
65imp 409 . . . 4 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → 𝐵 ∈ dom card)
7 cardid2 9376 . . . 4 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
86, 7syl 17 . . 3 (((card‘𝐴) = (card‘𝐵) ∧ ¬ (card‘𝐴) = ∅) → (card‘𝐵) ≈ 𝐵)
9 breq2 5062 . . . . 5 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) ↔ 𝐴 ≈ (card‘𝐵)))
10 entr 8555 . . . . . 6 ((𝐴 ≈ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴𝐵)
1110ex 415 . . . . 5 (𝐴 ≈ (card‘𝐵) → ((card‘𝐵) ≈ 𝐵𝐴𝐵))
129, 11syl6bi 255 . . . 4 ((card‘𝐴) = (card‘𝐵) → (𝐴 ≈ (card‘𝐴) → ((card‘𝐵) ≈ 𝐵𝐴𝐵)))
13 cardid2 9376 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
14 ndmfv 6694 . . . . . 6 𝐴 ∈ dom card → (card‘𝐴) = ∅)
1513, 14nsyl4 161 . . . . 5 (¬ (card‘𝐴) = ∅ → (card‘𝐴) ≈ 𝐴)
1615ensymd 8554 . . . 4 (¬ (card‘𝐴) = ∅ → 𝐴 ≈ (card‘𝐴))
1712, 16impel 508 . . 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 398   = wceq 1533  wcel 2110  wne 3016  c0 4290   class class class wbr 5058  dom cdm 5549  cfv 6349  cen 8500  cardccrd 9358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-er 8283  df-en 8504  df-card 9362
This theorem is referenced by:  card1  9391
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