Theorem tskwe 9926

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 5368	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		rabexg 5323	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V)
3		incom 4196	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
4		inex1g 5311	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) ∈ V)
5	3, 4	eqeltrrid 2837	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ V)
6		inss1 4223	. . . . . . . . . . 11 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ On
7	6	sseli 3973	. . . . . . . . . 10 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ On)
8		onelon 6377	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
9	8	ancoms 459	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ On) → 𝑦 ∈ On)
10	7, 9	sylan2 593	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ On)
11		onelss 6394	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
12	11	impcom 408	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ On) → 𝑦 ⊆ 𝑧)
13	7, 12	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ⊆ 𝑧)
14		inss2 4224	. . . . . . . . . . . . . . . . 17 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
15	14	sseli 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
16		breq1 5143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
17	16	elrab 3678	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
18	15, 17	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
19	18	simpld 495	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ 𝒫 𝐴)
20	19	elpwid 4604	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ⊆ 𝐴)
21	20	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ⊆ 𝐴)
22	13, 21	sstrd 3987	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ⊆ 𝐴)
23		velpw 4600	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
24	22, 23	sylibr 233	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25		vex 3476	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
26		ssdomg 8978	. . . . . . . . . . . 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 496	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ≺ 𝐴)
29	28	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ≺ 𝐴)
30		domsdomtr 9094	. . . . . . . . . . 11 ⊢ ((𝑦 ≼ 𝑧 ∧ 𝑧 ≺ 𝐴) → 𝑦 ≺ 𝐴)
31	27, 29, 30	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ≺ 𝐴)
32		breq1 5143	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
33	32	elrab 3678	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ 𝑦 ≺ 𝐴))
34	24, 31, 33	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
35	10, 34	elind 4189	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
36	35	gen2 1798	. . . . . . 7 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
37		dftr2 5259	. . . . . . 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 230	. . . . . 6 ⊢ Tr (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
39		ordon 7746	. . . . . 6 ⊢ Ord On
40		trssord 6369	. . . . . 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 1461	. . . . 5 ⊢ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
42		elong 6360	. . . . 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 257	. . . 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 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
46		simpr 485	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴)
47	14, 46	sstrid 3988	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴)
48		ssdomg 8978	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴))
49	48	adantr 481	. . . 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 6370	. . . . 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 1133	. . . . . 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 5336	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴))
55	54	adantr 481	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴)
57	56	3adant3 1132	. . . . . . 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 1138	. . . . . . 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 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥On
60		nfrab1 3450	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
61	59, 60	nfin 4211	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
62		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑥𝒫 𝐴
63		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≺
64		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
65	61, 63, 64	nfbr 5187	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴
66		breq1 5143	. . . . . . . 8 ⊢ (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑥 ≺ 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴))
67	61, 62, 65, 66	elrabf 3674	. . . . . . 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 583	. . . . . 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 4189	. . . . 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 1121	. . . 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 197	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴)
72		bren2 8961	. . 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 583	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≈ 𝐴)
74		isnumi 9922	. 2 ⊢ (((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On ∧ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≈ 𝐴) → 𝐴 ∈ dom card)
75	45, 73, 74	syl2anc 584	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → 𝐴 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087  ∀wal 1539   ∈ wcel 2106  {crab 3431  Vcvv 3472   ∩ cin 3942   ⊆ wss 3943  𝒫 cpw 4595   class class class wbr 5140  Tr wtr 5257  dom cdm 5668  Ord word 6351  Oncon0 6352   ≈ cen 8918   ≼ cdom 8919   ≺ csdm 8920  cardccrd 9911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pow 5355  ax-pr 5419  ax-un 7707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4943  df-br 5141  df-opab 5203  df-mpt 5224  df-tr 5258  df-id 5566  df-eprel 5572  df-po 5580  df-so 5581  df-fr 5623  df-we 5625  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-res 5680  df-ima 5681  df-ord 6355  df-on 6356  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-er 8685  df-en 8922  df-dom 8923  df-sdom 8924  df-card 9915

This theorem is referenced by:  tskwe2  10749  grothac  10806