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Theorem cardsdomel 9391
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 9370 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
21ensymd 8547 . . . . . 6 (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3 sdomentr 8639 . . . . . 6 ((𝐴𝐵𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
42, 3sylan2 595 . . . . 5 ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5 ssdomg 8542 . . . . . . . 8 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6 cardon 9361 . . . . . . . . 9 (card‘𝐵) ∈ On
7 domtriord 8651 . . . . . . . . 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 242 . . . . . . 7 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → ¬ 𝐴 ≺ (card‘𝐵)))
109con2d 136 . . . . . 6 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → ¬ (card‘𝐵) ⊆ 𝐴))
11 ontri1 6203 . . . . . . . 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 358 . . . . . 6 (𝐴 ∈ On → (𝐴 ∈ (card‘𝐵) ↔ ¬ (card‘𝐵) ⊆ 𝐴))
1410, 13sylibrd 262 . . . . 5 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → 𝐴 ∈ (card‘𝐵)))
154, 14syl5 34 . . . 4 (𝐴 ∈ On → ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ∈ (card‘𝐵)))
1615expcomd 420 . . 3 (𝐴 ∈ On → (𝐵 ∈ dom card → (𝐴𝐵𝐴 ∈ (card‘𝐵))))
1716imp 410 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
18 cardsdomelir 9390 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
1917, 18impbid1 228 1 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2114  wss 3908   class class class wbr 5042  dom cdm 5532  Oncon0 6169  cfv 6334  cen 8493  cdom 8494  csdm 8495  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ord 6172  df-on 6173  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-card 9356
This theorem is referenced by:  iscard  9392  cardval2  9408  infxpenlem  9428  alephnbtwn  9486  alephnbtwn2  9487  alephord2  9491  alephsdom  9501  pwsdompw  9615  inaprc  10247
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