Theorem cardsdomel 10043

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 10022	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2	1	ensymd 9065	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3		sdomentr 9177	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
4	2, 3	sylan2 592	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5		ssdomg 9060	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6		cardon 10013	. . . . . . . . 9 ⊢ (card‘𝐵) ∈ On
7		domtriord 9189	. . . . . . . . 9 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
8	6, 7	mpan 689	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
9	5, 8	sylibd 239	. . . . . . 7 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → ¬ 𝐴 ≺ (card‘𝐵)))
10	9	con2d 134	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → ¬ (card‘𝐵) ⊆ 𝐴))
11		ontri1 6429	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
12	6, 11	mpan 689	. . . . . . 7 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	12	con2bid 354	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ∈ (card‘𝐵) ↔ ¬ (card‘𝐵) ⊆ 𝐴))
14	10, 13	sylibrd 259	. . . . 5 ⊢ (𝐴 ∈ On → (𝐴 ≺ (card‘𝐵) → 𝐴 ∈ (card‘𝐵)))
15	4, 14	syl5 34	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ≺ 𝐵 ∧ 𝐵 ∈ dom card) → 𝐴 ∈ (card‘𝐵)))
16	15	expcomd 416	. . 3 ⊢ (𝐴 ∈ On → (𝐵 ∈ dom card → (𝐴 ≺ 𝐵 → 𝐴 ∈ (card‘𝐵))))
17	16	imp 406	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → 𝐴 ∈ (card‘𝐵)))
18		cardsdomelir 10042	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)
19	17, 18	impbid1 225	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 ↔ 𝐴 ∈ (card‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108   ⊆ wss 3976   class class class wbr 5166  dom cdm 5700  Oncon0 6395  ‘cfv 6573   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-card 10008

This theorem is referenced by:  iscard  10044  cardval2  10060  infxpenlem  10082  alephnbtwn  10140  alephnbtwn2  10141  alephord2  10145  alephsdom  10155  pwsdompw  10272  inaprc  10905