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Theorem cardsdomel 8657
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 8636 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
21ensymd 7867 . . . . . 6 (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3 sdomentr 7953 . . . . . 6 ((𝐴𝐵𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
42, 3sylan2 489 . . . . 5 ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5 ssdomg 7861 . . . . . . . 8 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6 cardon 8627 . . . . . . . . 9 (card‘𝐵) ∈ On
7 domtriord 7965 . . . . . . . . 9 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
86, 7mpan 701 . . . . . . . 8 (𝐴 ∈ On → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
95, 8sylibd 227 . . . . . . 7 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → ¬ 𝐴 ≺ (card‘𝐵)))
109con2d 127 . . . . . 6 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → ¬ (card‘𝐵) ⊆ 𝐴))
11 ontri1 5657 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
126, 11mpan 701 . . . . . . 7 (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1312con2bid 342 . . . . . 6 (𝐴 ∈ On → (𝐴 ∈ (card‘𝐵) ↔ ¬ (card‘𝐵) ⊆ 𝐴))
1410, 13sylibrd 247 . . . . 5 (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → 𝐴 ∈ (card‘𝐵)))
154, 14syl5 33 . . . 4 (𝐴 ∈ On → ((𝐴𝐵𝐵 ∈ dom card) → 𝐴 ∈ (card‘𝐵)))
1615expcomd 452 . . 3 (𝐴 ∈ On → (𝐵 ∈ dom card → (𝐴𝐵𝐴 ∈ (card‘𝐵))))
1716imp 443 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
18 cardsdomelir 8656 . 2 (𝐴 ∈ (card‘𝐵) → 𝐴𝐵)
1917, 18impbid1 213 1 ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴𝐵𝐴 ∈ (card‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wcel 1976  wss 3536   class class class wbr 4574  dom cdm 5025  Oncon0 5623  cfv 5787  cen 7812  cdom 7813  csdm 7814  cardccrd 8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-ord 5626  df-on 5627  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-card 8622
This theorem is referenced by:  iscard  8658  cardval2  8674  infxpenlem  8693  alephnbtwn  8751  alephnbtwn2  8752  alephord2  8756  alephsdom  8766  pwsdompw  8883  inaprc  9511
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