Theorem dfss4st 3256

Description: Subclass defined in terms of class difference. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
dfss4st	⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem dfss4st

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2160	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	stbid 783	. . 3 ⊢ (𝑥 = 𝑦 → (STAB 𝑥 ∈ 𝐴 ↔ STAB 𝑦 ∈ 𝐴))
3	2	cbvalv 1854	. 2 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 ↔ ∀𝑦STAB 𝑦 ∈ 𝐴)
4		nfa1 1489	. . . . 5 ⊢ Ⅎ𝑦∀𝑦STAB 𝑦 ∈ 𝐴
5		nfcv 2240	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∖ (𝐵 ∖ 𝐴))
6		nfcv 2240	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∩ 𝐴)
7		eldif 3030	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)))
8		elin 3206	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
9		abai 530	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
10	8, 9	bitri 183	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
11		imanst 785	. . . . . . . . . 10 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
12	11	anbi2d 455	. . . . . . . . 9 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
13	10, 12	syl5bb 191	. . . . . . . 8 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
14		eldif 3030	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
15	14	notbii 635	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
16	15	anbi2i 448	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
17	13, 16	syl6rbbr 198	. . . . . . 7 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
18	7, 17	syl5bb 191	. . . . . 6 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
19	18	sps 1485	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
20	4, 5, 6, 19	eqrd 3065	. . . 4 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴))
21	20	eqeq1d 2108	. . 3 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴))
22		sseqin2 3242	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
23	21, 22	syl6rbbr 198	. 2 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴))
24	3, 23	sylbi 120	1 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  STAB wstab 781  ∀wal 1297   = wceq 1299   ∈ wcel 1448   ∖ cdif 3018   ∩ cin 3020   ⊆ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-stab 782  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-in 3027  df-ss 3034

This theorem is referenced by:  sbthlemi3  6775