Theorem cardsdomel 9873

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 9852	. . . . . . 7 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
2	1	ensymd 8933	. . . . . 6 ⊢ (𝐵 ∈ dom card → 𝐵 ≈ (card‘𝐵))
3		sdomentr 9030	. . . . . 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 8928	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((card‘𝐵) ⊆ 𝐴 → (card‘𝐵) ≼ 𝐴))
6		cardon 9843	. . . . . . . . 9 ⊢ (card‘𝐵) ∈ On
7		domtriord 9042	. . . . . . . . 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 6346	. . . . . . . 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 9872	. 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 2111   ⊆ wss 3897   class class class wbr 5093  dom cdm 5619  Oncon0 6312  ‘cfv 6487   ≈ cen 8872   ≼ cdom 8873   ≺ csdm 8874  cardccrd 9834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6315  df-on 6316  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-card 9838

This theorem is referenced by:  iscard  9874  cardval2  9890  infxpenlem  9910  alephnbtwn  9968  alephnbtwn2  9969  alephord2  9973  alephsdom  9983  pwsdompw  10100  inaprc  10733