Theorem cardsdomel 9959

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) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵)))

Proof of Theorem cardsdomel

Step	Hyp	Ref	Expression
1		cardid2 9938	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2	1	ensymd 9001	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3		sdomentr 9098	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
4	2, 3	sylan2 604	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5		ssdomg 8996	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6		cardon 9929	. . . . . . . . 9 ⊢ (card‘𝐵) ∈ On
7		domtriord 9110	. . . . . . . . 9 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
8	6, 7	mpan 702	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
9	5, 8	sylibd 242	. . . . . . 7 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → ¬ 𝐴 ≺ (card‘𝐵)))
10	9	con2d 135	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → ¬ (card‘𝐵) ⊆ 𝐴))
11		ontri1 6395	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
12	6, 11	mpan 702	. . . . . . 7 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	12	con2bid 357	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ∈ (card‘𝐵) ↔ ¬ (card‘𝐵) ⊆ 𝐴))
14	10, 13	sylibrd 262	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → 𝐴 ∈ (card‘𝐵)))
15	4, 14	syl5 35	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ≺ 𝐵 ∧ 𝐵 ∈ dom card) → 𝐴 ∈ (card‘𝐵)))
16	15	expcomd 421	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ dom card → (𝐴 ≺ 𝐵 → 𝐴 ∈ (card‘𝐵))))
17	16	imp 411	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → 𝐴 ∈ (card‘𝐵)))
18		cardsdomelir 9958	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)
19	17, 18	impbid1 228	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2141   ⊆ wss 3904   class class class wbr 5108  dom cdm 5661  Oncon0 6360  ‘cfv 6536   ≈ cen 8939   ≼ cdom 8940   ≺ csdm 8941  cardccrd 9920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-card 9924

This theorem is referenced by:  iscard  9960  cardval2  9976  infxpenlem  9996  alephnbtwn  10054  alephnbtwn2  10055  alephord2  10059  alephsdom  10069  pwsdompw  10185  inaprc  10820