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Theorem cardne 8751
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)

Proof of Theorem cardne
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6187 . 2 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2 cardon 8730 . . . . . . . . . 10 (card‘𝐵) ∈ On
32oneli 5804 . . . . . . . . 9 (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4 breq1 4626 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
54onintss 5744 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (card‘𝐵) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
76adantl 482 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
8 cardval3 8738 . . . . . . . . 9 (𝐵 ∈ dom card → (card‘𝐵) = {𝑥 ∈ On ∣ 𝑥𝐵})
98sseq1d 3617 . . . . . . . 8 (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
109adantr 481 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
117, 10sylibrd 249 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → (card‘𝐵) ⊆ 𝐴))
12 ontri1 5726 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
132, 3, 12sylancr 694 . . . . . . 7 (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1413adantl 482 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1511, 14sylibd 229 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
1615con2d 129 . . . 4 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
1716ex 450 . . 3 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)))
1817pm2.43d 53 . 2 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
191, 18mpcom 38 1 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wcel 1987  {crab 2912  wss 3560   cint 4447   class class class wbr 4623  dom cdm 5084  Oncon0 5692  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-fv 5865  df-en 7916  df-card 8725
This theorem is referenced by:  carden2b  8753  cardlim  8758  cardsdomelir  8759
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