Theorem dfss4st 3355

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 2227	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	stbid 822	. . 3 ⊢ (𝑥 = 𝑦 → (STAB 𝑥 ∈ 𝐴 ↔ STAB 𝑦 ∈ 𝐴))
3	2	cbvalv 1905	. 2 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 ↔ ∀𝑦STAB 𝑦 ∈ 𝐴)
4		sseqin2 3341	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
5		nfa1 1529	. . . . 5 ⊢ Ⅎ𝑦∀𝑦STAB 𝑦 ∈ 𝐴
6		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∖ (𝐵 ∖ 𝐴))
7		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∩ 𝐴)
8		eldif 3125	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)))
9		eldif 3125	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
10	9	notbii 658	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
11	10	anbi2i 453	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
12		elin 3305	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
13		abai 550	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
14	12, 13	bitri 183	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)))
15		imanst 878	. . . . . . . . . 10 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
16	15	anbi2d 460	. . . . . . . . 9 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
17	14, 16	syl5bb 191	. . . . . . . 8 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))))
18	11, 17	bitr4id 198	. . . . . . 7 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
19	8, 18	syl5bb 191	. . . . . 6 ⊢ (STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
20	19	sps 1525	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
21	5, 6, 7, 20	eqrd 3160	. . . 4 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴))
22	21	eqeq1d 2174	. . 3 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴))
23	4, 22	bitr4id 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 820  ∀wal 1341   = wceq 1343   ∈ wcel 2136   ∖ cdif 3113   ∩ cin 3115   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-stab 821  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129

This theorem is referenced by:  sbthlemi3  6924