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Theorem carden2b 8778
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8777 are meant to replace carden 9358 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 8776 . . . . 5 ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2 ennum 8758 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
32biimpa 501 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4 cardid2 8764 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
53, 4syl 17 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6 ensym 7990 . . . . . . 7 (𝐴𝐵𝐵𝐴)
76adantr 481 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵𝐴)
8 entr 7993 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵𝐴) → (card‘𝐵) ≈ 𝐴)
95, 7, 8syl2anc 692 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
101, 9nsyl3 133 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11 cardon 8755 . . . . 5 (card‘𝐴) ∈ On
12 cardon 8755 . . . . 5 (card‘𝐵) ∈ On
13 ontri1 5745 . . . . 5 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
1411, 12, 13mp2an 707 . . . 4 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
1510, 14sylibr 224 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16 cardne 8776 . . . . 5 ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17 cardid2 8764 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
19 entr 7993 . . . . . 6 (((card‘𝐴) ≈ 𝐴𝐴𝐵) → (card‘𝐴) ≈ 𝐵)
2017, 18, 19syl2anr 495 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
2116, 20nsyl3 133 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22 ontri1 5745 . . . . 5 (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
2312, 11, 22mp2an 707 . . . 4 ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
2421, 23sylibr 224 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
2515, 24eqssd 3612 . 2 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26 ndmfv 6205 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
2726adantl 482 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
282notbid 308 . . . . 5 (𝐴𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
2928biimpa 501 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30 ndmfv 6205 . . . 4 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3129, 30syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
3227, 31eqtr4d 2657 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
3325, 32pm2.61dan 831 1 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wss 3567  c0 3907   class class class wbr 4644  dom cdm 5104  Oncon0 5711  cfv 5876  cen 7937  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-er 7727  df-en 7941  df-card 8750
This theorem is referenced by:  card1  8779  carddom2  8788  cardennn  8794  cardsucinf  8795  pm54.43lem  8810  nnacda  9008  ficardun  9009  ackbij1lem5  9031  ackbij1lem8  9034  ackbij1lem9  9035  ackbij2lem2  9047  carden  9358  r1tskina  9589  cardfz  12752
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