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Theorem cardsdomel 9965
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. (Contributed by Mario Carneiro, 15-Jan-2013.) (Revised by Mario Carneiro, 4-Jun-2015.)
Assertion
Ref Expression
cardsdomel ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))

Proof of Theorem cardsdomel
StepHypRef Expression
1 cardid2 9944 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
21ensymd 8997 . . . . . 6 (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3 sdomentr 9107 . . . . . 6 ((𝐴𝐵𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
42, 3sylan2 594 . . . . 5 ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5 ssdomg 8992 . . . . . . . 8 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6 cardon 9935 . . . . . . . . 9 (card‘𝐵) ∈ On
7 domtriord 9119 . . . . . . . . 9 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
86, 7mpan 689 . . . . . . . 8 (𝐴 ∈ On → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
95, 8sylibd 238 . . . . . . 7 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → ¬ 𝐴 ≺ (card‘𝐵)))
109con2d 134 . . . . . 6 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → ¬ (card‘𝐵) ⊆ 𝐴))
11 ontri1 6395 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
126, 11mpan 689 . . . . . . 7 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1312con2bid 355 . . . . . 6 (𝐴 ∈ On → (𝐴 ∈ (card‘𝐵) ↔ ¬ (card‘𝐵) ⊆ 𝐴))
1410, 13sylibrd 259 . . . . 5 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → 𝐴 ∈ (card‘𝐵)))
154, 14syl5 34 . . . 4 (𝐴 ∈ On → ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ∈ (card‘𝐵)))
1615expcomd 418 . . 3 (𝐴 ∈ On → (𝐵 ∈ dom card → (𝐴𝐵𝐴 ∈ (card‘𝐵))))
1716imp 408 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
18 cardsdomelir 9964 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
1917, 18impbid1 224 1 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wcel 2107  wss 3947   class class class wbr 5147  dom cdm 5675  Oncon0 6361  cfv 6540  cen 8932  cdom 8933  csdm 8934  cardccrd 9926
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-card 9930
This theorem is referenced by:  iscard  9966  cardval2  9982  infxpenlem  10004  alephnbtwn  10062  alephnbtwn2  10063  alephord2  10067  alephsdom  10077  pwsdompw  10195  inaprc  10827
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