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Theorem cardsdomel 9997
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 9976 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
21ensymd 9024 . . . . . 6 (𝐡 ∈ dom card β†’ 𝐡 β‰ˆ (cardβ€˜π΅))
3 sdomentr 9134 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 β‰ˆ (cardβ€˜π΅)) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
42, 3sylan2 591 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 ∈ dom card) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
5 ssdomg 9019 . . . . . . . 8 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 β†’ (cardβ€˜π΅) β‰Ό 𝐴))
6 cardon 9967 . . . . . . . . 9 (cardβ€˜π΅) ∈ On
7 domtriord 9146 . . . . . . . . 9 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) β‰Ό 𝐴 ↔ Β¬ 𝐴 β‰Ί (cardβ€˜π΅)))
86, 7mpan 688 . . . . . . . 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 6398 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
126, 11mpan 688 . . . . . . 7 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1312con2bid 353 . . . . . 6 (𝐴 ∈ On β†’ (𝐴 ∈ (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) βŠ† 𝐴))
1410, 13sylibrd 258 . . . . 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 415 . . 3 (𝐴 ∈ On β†’ (𝐡 ∈ dom card β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅))))
1716imp 405 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅)))
18 cardsdomelir 9996 . 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 394   ∈ wcel 2098   βŠ† wss 3939   class class class wbr 5143  dom cdm 5672  Oncon0 6364  β€˜cfv 6543   β‰ˆ cen 8959   β‰Ό cdom 8960   β‰Ί csdm 8961  cardccrd 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-card 9962
This theorem is referenced by:  iscard  9998  cardval2  10014  infxpenlem  10036  alephnbtwn  10094  alephnbtwn2  10095  alephord2  10099  alephsdom  10109  pwsdompw  10227  inaprc  10859
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