Theorem tskwe 9902

Description: A Tarski set is well-orderable. (Contributed by Mario Carneiro, 19-Apr-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
tskwe	⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem tskwe

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwexg 5332	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		rabexg 5290	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V)
3		incom 4159	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
4		inex1g 5272	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) ∈ V)
5	3, 4	eqeltrrid 2866	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ V)
6		inss1 4186	. . . . . . . . . . 11 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ On
7	6	sseli 3930	. . . . . . . . . 10 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ On)
8		onelon 6366	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
9	8	ancoms 462	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ On) → 𝑦 ∈ On)
10	7, 9	sylan2 602	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ On)
11		onelss 6383	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
12	11	impcom 411	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ On) → 𝑦 ⊆ 𝑧)
13	7, 12	sylan2 602	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ⊆ 𝑧)
14		inss2 4187	. . . . . . . . . . . . . . . . 17 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
15	14	sseli 3930	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
16		breq1 5100	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
17	16	elrab 3649	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
18	15, 17	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
19	18	simpld 498	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ 𝒫 𝐴)
20	19	elpwid 4561	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ⊆ 𝐴)
21	20	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ⊆ 𝐴)
22	13, 21	sstrd 3944	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ⊆ 𝐴)
23		velpw 4557	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
24	22, 23	sylibr 236	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
26		ssdomg 8975	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → (𝑦 ⊆ 𝑧 → 𝑦 ≼ 𝑧))
27	25, 13, 26	mpsyl 68	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ≼ 𝑧)
28	18	simprd 499	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ≺ 𝐴)
29	28	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ≺ 𝐴)
30		domsdomtr 9078	. . . . . . . . . . 11 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
31	27, 29, 30	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ≺ 𝐴)
32		breq1 5100	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
33	32	elrab 3649	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ≺ 𝐴))
34	24, 31, 33	sylanbrc 592	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
35	10, 34	elind 4150	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
36	35	gen2 1815	. . . . . . 7 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
37		dftr2 5206	. . . . . . 7 ⊢ (Tr (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ↔ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})))
38	36, 37	mpbir 233	. . . . . 6 ⊢ Tr (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
39		ordon 7755	. . . . . 6 ⊢ Ord On
40		trssord 6358	. . . . . 6 ⊢ ((Tr (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ On ∧ Ord On) → Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
41	38, 6, 39, 40	mp3an 1481	. . . . 5 ⊢ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
42		elong 6349	. . . . 5 ⊢ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ V → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On ↔ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})))
43	41, 42	mpbiri 260	. . . 4 ⊢ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
44	1, 2, 5, 43	4syl 19	. . 3 ⊢ (𝐴 ∈ 𝑉 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
45	44	adantr 484	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
46		simpr 488	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴)
47	14, 46	sstrid 3945	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴)
48		ssdomg 8975	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴))
49	48	adantr 484	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴))
50	47, 49	mpd 15	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴)
51		ordirr 6359	. . . . 5 ⊢ (Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
52	41, 51	mp1i 13	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
53	44	3ad2ant1 1145	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
54		elpw2g 5286	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴))
55	54	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴))
56	47, 55	mpbird 259	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴)
57	56	3adant3 1144	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴)
58		simp3 1150	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴)
59		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥On
60		nfrab1 3433	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
61	59, 60	nfin 4174	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
62		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝒫 𝐴
63		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≺
64		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
65	61, 63, 64	nfbr 5144	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴
66		breq1 5100	. . . . . . . 8 ⊢ (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑥 ≺ 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴))
67	61, 62, 65, 66	elrabf 3646	. . . . . . 7 ⊢ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴))
68	57, 58, 67	sylanbrc 592	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
69	53, 68	elind 4150	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴 ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
70	69	3expia 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})))
71	52, 70	mtod 200	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴)
72		bren2 8958	. . 3 ⊢ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≈ 𝐴 ↔ ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴 ∧ ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴))
73	50, 71, 72	sylanbrc 592	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≈ 𝐴)
74		isnumi 9898	. 2 ⊢ (((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
75	45, 73, 74	syl2anc 593	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097  ∀wal 1557   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4552   class class class wbr 5097  Tr wtr 5204  dom cdm 5643  Ord word 6340  Oncon0 6341   ≈ cen 8918   ≼ cdom 8919   ≺ csdm 8920  cardccrd 9887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-ord 6344  df-on 6345  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-card 9891

This theorem is referenced by:  tskwe2  10725  grothac  10782