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Theorem cardsdomel 9969
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 ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ί 𝐡 ↔ 𝐴 ∈ (cardβ€˜π΅)))
Proof of Theorem cardsdomel
StepHypRef Expression
1 cardid2 9948 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
21ensymd 9001 . . . . . 6 (𝐡 ∈ dom card β†’ 𝐡 β‰ˆ (cardβ€˜π΅))
3 sdomentr 9111 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 β‰ˆ (cardβ€˜π΅)) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
42, 3sylan2 594 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 ∈ dom card) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
5 ssdomg 8996 . . . . . . . 8 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 β†’ (cardβ€˜π΅) β‰Ό 𝐴))
6 cardon 9939 . . . . . . . . 9 (cardβ€˜π΅) ∈ On
7 domtriord 9123 . . . . . . . . 9 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) β‰Ό 𝐴 ↔ Β¬ 𝐴 β‰Ί (cardβ€˜π΅)))
86, 7mpan 689 . . . . . . . 8 (𝐴 ∈ On β†’ ((cardβ€˜π΅) β‰Ό 𝐴 ↔ Β¬ 𝐴 β‰Ί (cardβ€˜π΅)))
95, 8sylibd 238 . . . . . . 7 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 β†’ Β¬ 𝐴 β‰Ί (cardβ€˜π΅)))
109con2d 134 . . . . . 6 (𝐴 ∈ On β†’ (𝐴 β‰Ί (cardβ€˜π΅) β†’ Β¬ (cardβ€˜π΅) βŠ† 𝐴))
11 ontri1 6399 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
126, 11mpan 689 . . . . . . 7 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1312con2bid 355 . . . . . 6 (𝐴 ∈ On β†’ (𝐴 ∈ (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) βŠ† 𝐴))
1410, 13sylibrd 259 . . . . 5 (𝐴 ∈ On β†’ (𝐴 β‰Ί (cardβ€˜π΅) β†’ 𝐴 ∈ (cardβ€˜π΅)))
154, 14syl5 34 . . . 4 (𝐴 ∈ On β†’ ((𝐴 β‰Ί 𝐡 ∧ 𝐡 ∈ dom card) β†’ 𝐴 ∈ (cardβ€˜π΅)))
1615expcomd 418 . . 3 (𝐴 ∈ On β†’ (𝐡 ∈ dom card β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅))))
1716imp 408 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅)))
18 cardsdomelir 9968 . 2 (𝐴 ∈ (cardβ€˜π΅) β†’ 𝐴 β‰Ί 𝐡)
1917, 18impbid1 224 1 ((𝐴 ∈ On ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ί 𝐡 ↔ 𝐴 ∈ (cardβ€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677  Oncon0 6365  β€˜cfv 6544   β‰ˆ cen 8936   β‰Ό cdom 8937   β‰Ί csdm 8938  cardccrd 9930
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9934
This theorem is referenced by:  iscard  9970  cardval2  9986  infxpenlem  10008  alephnbtwn  10066  alephnbtwn2  10067  alephord2  10071  alephsdom  10081  pwsdompw  10199  inaprc  10831
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