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Theorem cardsdomel 9971
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 9950 . . . . . . 7 (𝐡 ∈ dom card β†’ (cardβ€˜π΅) β‰ˆ 𝐡)
21ensymd 9003 . . . . . 6 (𝐡 ∈ dom card β†’ 𝐡 β‰ˆ (cardβ€˜π΅))
3 sdomentr 9113 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 β‰ˆ (cardβ€˜π΅)) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
42, 3sylan2 592 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝐡 ∈ dom card) β†’ 𝐴 β‰Ί (cardβ€˜π΅))
5 ssdomg 8998 . . . . . . . 8 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 β†’ (cardβ€˜π΅) β‰Ό 𝐴))
6 cardon 9941 . . . . . . . . 9 (cardβ€˜π΅) ∈ On
7 domtriord 9125 . . . . . . . . 9 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) β‰Ό 𝐴 ↔ Β¬ 𝐴 β‰Ί (cardβ€˜π΅)))
86, 7mpan 687 . . . . . . . 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 6392 . . . . . . . 8 (((cardβ€˜π΅) ∈ On ∧ 𝐴 ∈ On) β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
126, 11mpan 687 . . . . . . 7 (𝐴 ∈ On β†’ ((cardβ€˜π΅) βŠ† 𝐴 ↔ Β¬ 𝐴 ∈ (cardβ€˜π΅)))
1312con2bid 354 . . . . . 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 416 . . 3 (𝐴 ∈ On β†’ (𝐡 ∈ dom card β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅))))
1716imp 406 . 2 ((𝐴 ∈ On ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ (cardβ€˜π΅)))
18 cardsdomelir 9970 . 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 395   ∈ wcel 2098   βŠ† wss 3943   class class class wbr 5141  dom cdm 5669  Oncon0 6358  β€˜cfv 6537   β‰ˆ cen 8938   β‰Ό cdom 8939   β‰Ί csdm 8940  cardccrd 9932
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-card 9936
This theorem is referenced by:  iscard  9972  cardval2  9988  infxpenlem  10010  alephnbtwn  10068  alephnbtwn2  10069  alephord2  10073  alephsdom  10083  pwsdompw  10201  inaprc  10833
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