Theorem dfss4st 3464

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 2299	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2	1	stbid 844	. . 3 ⊢ (𝑥 = 𝑦 → (STAB 𝑥 ∈ 𝐴 ↔ STAB 𝑦 ∈ 𝐴))
3	2	cbvalv 1973	. 2 ⊢ (∀𝑥STAB 𝑥 ∈ 𝐴 ↔ ∀𝑦STAB 𝑦 ∈ 𝐴)
4		sseqin2 3450	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐵 ∩ 𝐴) = 𝐴)
5		nfa1 1594	. . . . 5 ⊢ Ⅎ𝑦∀𝑦STAB 𝑦 ∈ 𝐴
6		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∖ (𝐵 ∖ 𝐴))
7		nfcv 2392	. . . . 5 ⊢ Ⅎ𝑦(𝐵 ∩ 𝐴)
8		eldif 3229	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)))
9		eldif 3229	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∖ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
10	9	notbii 678	. . . . . . . . 9 ⊢ (¬ 𝑦 ∈ (𝐵 ∖ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴))
11	10	anbi2i 461	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ (𝐵 ∖ 𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
12		elin 3412	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐵 ∩ 𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴))
13		abai 566	. . . . . . . . . 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 900	. . . . . . . . . 10 ⊢ (STAB 𝑦 ∈ 𝐴 → ((𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴) ↔ ¬ (𝑦 ∈ 𝐵 ∧ ¬ 𝑦 ∈ 𝐴)))
16	15	anbi2d 468	. . . . . . . . 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 1590	. . . . 5 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝑦 ∈ (𝐵 ∖ (𝐵 ∖ 𝐴)) ↔ 𝑦 ∈ (𝐵 ∩ 𝐴)))
21	5, 6, 7, 20	eqrd 3266	. . . 4 ⊢ (∀𝑦STAB 𝑦 ∈ 𝐴 → (𝐵 ∖ (𝐵 ∖ 𝐴)) = (𝐵 ∩ 𝐴))
22	21	eqeq1d 2247	. . 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 842  ∀wal 1400   = wceq 1402   ∈ wcel 2209   ∖ cdif 3217   ∩ cin 3219   ⊆ wss 3220

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-stab 843  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233

This theorem is referenced by:  sbthlemi3  7270