Theorem cardsdomel 9927

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 9906	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2	1	ensymd 8976	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3		sdomentr 9075	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ (card‘𝐵)) → 𝐴 ≺ (card‘𝐵))
4	2, 3	sylan2 593	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ∈ dom card) → 𝐴 ≺ (card‘𝐵))
5		ssdomg 8971	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6		cardon 9897	. . . . . . . . 9 ⊢ (card‘𝐵) ∈ On
7		domtriord 9087	. . . . . . . . 9 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (card‘𝐵)))
8	6, 7	mpan 690	. . . . . . . 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 6366	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
12	6, 11	mpan 690	. . . . . . 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 9926	. 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 2109   ⊆ wss 3914   class class class wbr 5107  dom cdm 5638  Oncon0 6332  ‘cfv 6511   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-card 9892

This theorem is referenced by:  iscard  9928  cardval2  9944  infxpenlem  9966  alephnbtwn  10024  alephnbtwn2  10025  alephord2  10029  alephsdom  10039  pwsdompw  10156  inaprc  10789