Theorem cardsdomel 3658

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93 (use cardsdom 3643 to obtain the exact proposition from this one).

Assertion

Ref	Expression
cardsdomel	⊢ (A ∈ On → (A ≺ B ↔ A ∈ (card ‘B)))

Proof of Theorem cardsdomel

Step	Hyp	Ref	Expression
1		ssdom2g 3312	. . . . 5 ⊢ (A ∈ On → ((card ‘B) ⊆ A → (card ‘B) ≼ A))
2		cardon 3634	. . . . . 6 ⊢ (card ‘B) ∈ On
3		ontri1 2232	. . . . . 6 ⊢ (((card ‘B) ∈ On ∧ A ∈ On) → ((card ‘B) ⊆ A ↔ ¬ A ∈ (card ‘B)))
4	2, 3	mpan 518	. . . . 5 ⊢ (A ∈ On → ((card ‘B) ⊆ A ↔ ¬ A ∈ (card ‘B)))
5		domtri 3644	. . . . . 6 ⊢ (((card ‘B) ∈ On ∧ A ∈ On) → ((card ‘B) ≼ A ↔ ¬ A ≺ (card ‘B)))
6	2, 5	mpan 518	. . . . 5 ⊢ (A ∈ On → ((card ‘B) ≼ A ↔ ¬ A ≺ (card ‘B)))
7	1, 4, 6	3imtr3d 420	. . . 4 ⊢ (A ∈ On → (¬ A ∈ (card ‘B) → ¬ A ≺ (card ‘B)))
8	7	a3d 70	. . 3 ⊢ (A ∈ On → (A ≺ (card ‘B) → A ∈ (card ‘B)))
9	2	onelss 2348	. . . . . . 7 ⊢ (A ∈ (card ‘B) → A ⊆ (card ‘B))
10		ssdom2g 3312	. . . . . . . 8 ⊢ ((card ‘B) ∈ On → (A ⊆ (card ‘B) → A ≼ (card ‘B)))
11	2, 10	ax-mp 6	. . . . . . 7 ⊢ (A ⊆ (card ‘B) → A ≼ (card ‘B))
12	9, 11	syl 12	. . . . . 6 ⊢ (A ∈ (card ‘B) → A ≼ (card ‘B))
13		cardcard 3655	. . . . . . . 8 ⊢ (card ‘(card ‘B)) = (card ‘B)
14	13	eleq2i 1153	. . . . . . 7 ⊢ (A ∈ (card ‘(card ‘B)) ↔ A ∈ (card ‘B))
15		cardne 3637	. . . . . . 7 ⊢ (A ∈ (card ‘(card ‘B)) → ¬ A ≈ (card ‘B))
16	14, 15	sylbir 176	. . . . . 6 ⊢ (A ∈ (card ‘B) → ¬ A ≈ (card ‘B))
17	12, 16	jca 236	. . . . 5 ⊢ (A ∈ (card ‘B) → (A ≼ (card ‘B) ∧ ¬ A ≈ (card ‘B)))
18		brsdom 3286	. . . . 5 ⊢ (A ≺ (card ‘B) ↔ (A ≼ (card ‘B) ∧ ¬ A ≈ (card ‘B)))
19	17, 18	sylibr 175	. . . 4 ⊢ (A ∈ (card ‘B) → A ≺ (card ‘B))
20	19	a1i 7	. . 3 ⊢ (A ∈ On → (A ∈ (card ‘B) → A ≺ (card ‘B)))
21	8, 20	impbid 397	. 2 ⊢ (A ∈ On → (A ≺ (card ‘B) ↔ A ∈ (card ‘B)))
22		sdomsdomcard 3654	. 2 ⊢ (A ≺ B ↔ A ≺ (card ‘B))
23	21, 22	syl5bb 410	1 ⊢ (A ∈ On → (A ≺ B ↔ A ∈ (card ‘B)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092   ⊆ wss 1487   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271   ≼ cdom 3272   ≺ csdm 3273  cardccrd 3620

This theorem is referenced by:  iscard 3659  cardval2 3661  alephnbtwn 3674  alephnbtwn2 3675  alephord2 3681

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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