Theorem dfss4st 3396

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 2257	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	stbid 833	. . 3 ⊢ (𝑥 = 𝑦 → (STAB 𝑥 ∈ 𝐴 ↔ STAB 𝑦 ∈ 𝐴))
3	2	cbvalv 1932	. 2 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 ↔ ∀𝑦STAB 𝑦 ∈ 𝐴)
4		sseqin2 3382	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
5		nfa1 1555	. . . . 5 ⊢ Ⅎ𝑦∀𝑦STAB 𝑦 ∈ 𝐴
6		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∖ (𝐵 ∖ 𝐴))
7		nfcv 2339	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∩ 𝐴)
8		eldif 3166	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)))
9		eldif 3166	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
10	9	notbii 669	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
11	10	anbi2i 457	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
12		elin 3346	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
13		abai 560	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
14	12, 13	bitri 184	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
15		imanst 889	. . . . . . . . . 10 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
16	15	anbi2d 464	. . . . . . . . 9 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
17	14, 16	bitrid 192	. . . . . . . 8 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
18	11, 17	bitr4id 199	. . . . . . 7 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
19	8, 18	bitrid 192	. . . . . 6 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
20	19	sps 1551	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
21	5, 6, 7, 20	eqrd 3201	. . . 4 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴))
22	21	eqeq1d 2205	. . 3 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴))
23	4, 22	bitr4id 199	. 2 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴))
24	3, 23	sylbi 121	1 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 → (𝐴 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  STAB wstab 831  ∀wal 1362   = wceq 1364   ∈ wcel 2167   ∖ cdif 3154   ∩ cin 3156   ⊆ wss 3157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-stab 832  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170

This theorem is referenced by:  sbthlemi3  7025