Theorem dfss4st 3309

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 2200	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	stbid 817	. . 3 ⊢ (𝑥 = 𝑦 → (STAB 𝑥 ∈ 𝐴 ↔ STAB 𝑦 ∈ 𝐴))
3	2	cbvalv 1889	. 2 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 ↔ ∀𝑦STAB 𝑦 ∈ 𝐴)
4		nfa1 1521	. . . . 5 ⊢ Ⅎ𝑦∀𝑦STAB 𝑦 ∈ 𝐴
5		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∖ (𝐵 ∖ 𝐴))
6		nfcv 2281	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∩ 𝐴)
7		eldif 3080	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)))
8		elin 3259	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
9		abai 549	. . . . . . . . . 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 873	. . . . . . . . . 10 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
12	11	anbi2d 459	. . . . . . . . 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 3080	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
15	14	notbii 657	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
16	15	anbi2i 452	. . . . . . . 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 1517	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
20	4, 5, 6, 19	eqrd 3115	. . . 4 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴))
21	20	eqeq1d 2148	. . 3 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → ((𝐵 ∖ (𝐵 ∖ 𝐴)) = 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐴))
22		sseqin2 3295	. . 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 815  ∀wal 1329   = wceq 1331   ∈ wcel 1480   ∖ cdif 3068   ∩ cin 3070   ⊆ wss 3071

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-stab 816  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084

This theorem is referenced by:  sbthlemi3  6847