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Theorem carden2b 8993
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 8992 are meant to replace carden 9575 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 8991 . . . . 5 ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2 ennum 8973 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
32biimpa 462 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4 cardid2 8979 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
53, 4syl 17 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6 ensym 8158 . . . . . . 7 (𝐴𝐵𝐵𝐴)
76adantr 466 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵𝐴)
8 entr 8161 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵𝐴) → (card‘𝐵) ≈ 𝐴)
95, 7, 8syl2anc 573 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
101, 9nsyl3 135 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11 cardon 8970 . . . . 5 (card‘𝐴) ∈ On
12 cardon 8970 . . . . 5 (card‘𝐵) ∈ On
13 ontri1 5900 . . . . 5 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
1411, 12, 13mp2an 672 . . . 4 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
1510, 14sylibr 224 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16 cardne 8991 . . . . 5 ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17 cardid2 8979 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
19 entr 8161 . . . . . 6 (((card‘𝐴) ≈ 𝐴𝐴𝐵) → (card‘𝐴) ≈ 𝐵)
2017, 18, 19syl2anr 584 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
2116, 20nsyl3 135 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22 ontri1 5900 . . . . 5 (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
2312, 11, 22mp2an 672 . . . 4 ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
2421, 23sylibr 224 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
2515, 24eqssd 3769 . 2 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26 ndmfv 6359 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
2726adantl 467 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
282notbid 307 . . . . 5 (𝐴𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
2928biimpa 462 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30 ndmfv 6359 . . . 4 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3129, 30syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
3227, 31eqtr4d 2808 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
3325, 32pm2.61dan 813 1 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wss 3723  c0 4063   class class class wbr 4786  dom cdm 5249  Oncon0 5866  cfv 6031  cen 8106  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-er 7896  df-en 8110  df-card 8965
This theorem is referenced by:  card1  8994  carddom2  9003  cardennn  9009  cardsucinf  9010  pm54.43lem  9025  nnacda  9225  ficardun  9226  ackbij1lem5  9248  ackbij1lem8  9251  ackbij1lem9  9252  ackbij2lem2  9264  carden  9575  r1tskina  9806  cardfz  12977
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