Theorem tskwe 9879

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 5328	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		rabexg 5287	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V)
3		incom 4168	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
4		inex1g 5269	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∩ On) ∈ V)
5	3, 4	eqeltrrid 2833	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ∈ V → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ V)
6		inss1 4196	. . . . . . . . . . 11 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ On
7	6	sseli 3939	. . . . . . . . . 10 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ On)
8		onelon 6345	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ 𝑦 ∈ 𝑧) → 𝑦 ∈ On)
9	8	ancoms 458	. . . . . . . . . 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 6362	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ On → (𝑦 ∈ 𝑧 → 𝑦 ⊆ 𝑧))
12	11	impcom 407	. . . . . . . . . . . . 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 4197	. . . . . . . . . . . . . . . . 17 ⊢ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
15	14	sseli 3939	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
16		breq1 5105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝑥 ≺ 𝐴 ↔ 𝑧 ≺ 𝐴))
17	16	elrab 3656	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
18	15, 17	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑧 ∈ 𝒫 𝐴 ∧ 𝑧 ≺ 𝐴))
19	18	simpld 494	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ∈ 𝒫 𝐴)
20	19	elpwid 4568	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ⊆ 𝐴)
21	20	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ⊆ 𝐴)
22	13, 21	sstrd 3954	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ⊆ 𝐴)
23		velpw 4564	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 𝐴 ↔ 𝑦 ⊆ 𝐴)
24	22, 23	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ 𝒫 𝐴)
25		vex 3448	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
26		ssdomg 8948	. . . . . . . . . . . 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 495	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → 𝑧 ≺ 𝐴)
29	28	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑧 ≺ 𝐴)
30		domsdomtr 9053	. . . . . . . . . . 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 5105	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ≺ 𝐴 ↔ 𝑦 ≺ 𝐴))
33	32	elrab 3656	. . . . . . . . . 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 4159	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
36	35	gen2 1796	. . . . . . 7 ⊢ ∀𝑦∀𝑧((𝑦 ∈ 𝑧 ∧ 𝑧 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})) → 𝑦 ∈ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}))
37		dftr2 5211	. . . . . . 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 231	. . . . . 6 ⊢ Tr (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
39		ordon 7733	. . . . . 6 ⊢ Ord On
40		trssord 6337	. . . . . 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 1463	. . . . 5 ⊢ Ord (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
42		elong 6328	. . . . 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 258	. . . 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 480	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ On)
46		simpr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴)
47	14, 46	sstrid 3955	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴)
48		ssdomg 8948	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴 → (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≼ 𝐴))
49	48	adantr 480	. . . 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 6338	. . . . 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 5283	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ((On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ∈ 𝒫 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ⊆ 𝐴))
55	54	adantr 480	. . . . . . . . 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 257	. . . . . . . 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 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥On
60		nfrab1 3423	. . . . . . . . 9 ⊢ Ⅎ𝑥{𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}
61	59, 60	nfin 4183	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴})
62		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑥𝒫 𝐴
63		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥 ≺
64		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐴
65	61, 63, 64	nfbr 5149	. . . . . . . 8 ⊢ Ⅎ𝑥(On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴
66		breq1 5105	. . . . . . . 8 ⊢ (𝑥 = (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) → (𝑥 ≺ 𝐴 ↔ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴))
67	61, 62, 65, 66	elrabf 3652	. . . . . . 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 4159	. . . . 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 198	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴} ⊆ 𝐴) → ¬ (On ∩ {𝑥 ∈ 𝒫 𝐴 ∣ 𝑥 ≺ 𝐴}) ≺ 𝐴)
72		bren2 8931	. . 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 9875	. 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 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   ∈ wcel 2109  {crab 3402  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559   class class class wbr 5102  Tr wtr 5209  dom cdm 5631  Ord word 6319  Oncon0 6320   ≈ cen 8892   ≼ cdom 8893   ≺ csdm 8894  cardccrd 9864

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-card 9868

This theorem is referenced by:  tskwe2  10702  grothac  10759