Theorem ssofss 4077

Description: Condition for subset when A is already known to be a subset. (Contributed by SF, 13-Jan-2015.)

Assertion

Ref	Expression
ssofss	⊢ (A ⊆ C → (A ⊆ B ↔ ∀x ∈ C (x ∈ A → x ∈ B)))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem ssofss

Step	Hyp	Ref	Expression
1		vex 2863	. . . . . . . 8 ⊢ x ∈ V
2	1	elcompl 3226	. . . . . . 7 ⊢ (x ∈ ∼ C ↔ ¬ x ∈ C)
3		ssel 3268	. . . . . . . 8 ⊢ (A ⊆ C → (x ∈ A → x ∈ C))
4	3	con3d 125	. . . . . . 7 ⊢ (A ⊆ C → (¬ x ∈ C → ¬ x ∈ A))
5	2, 4	syl5bi 208	. . . . . 6 ⊢ (A ⊆ C → (x ∈ ∼ C → ¬ x ∈ A))
6	5	imp 418	. . . . 5 ⊢ ((A ⊆ C ∧ x ∈ ∼ C) → ¬ x ∈ A)
7	6	pm2.21d 98	. . . 4 ⊢ ((A ⊆ C ∧ x ∈ ∼ C) → (x ∈ A → x ∈ B))
8	7	ralrimiva 2698	. . 3 ⊢ (A ⊆ C → ∀x ∈ ∼ C(x ∈ A → x ∈ B))
9	8	biantrud 493	. 2 ⊢ (A ⊆ C → (∀x ∈ C (x ∈ A → x ∈ B) ↔ (∀x ∈ C (x ∈ A → x ∈ B) ∧ ∀x ∈ ∼ C(x ∈ A → x ∈ B))))
10		ralv 2873	. . . 4 ⊢ (∀x ∈ V (x ∈ A → x ∈ B) ↔ ∀x(x ∈ A → x ∈ B))
11		uncompl 4075	. . . . 5 ⊢ (C ∪ ∼ C) = V
12	11	raleqi 2812	. . . 4 ⊢ (∀x ∈ (C ∪ ∼ C)(x ∈ A → x ∈ B) ↔ ∀x ∈ V (x ∈ A → x ∈ B))
13		dfss2 3263	. . . 4 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
14	10, 12, 13	3bitr4ri 269	. . 3 ⊢ (A ⊆ B ↔ ∀x ∈ (C ∪ ∼ C)(x ∈ A → x ∈ B))
15		ralunb 3445	. . 3 ⊢ (∀x ∈ (C ∪ ∼ C)(x ∈ A → x ∈ B) ↔ (∀x ∈ C (x ∈ A → x ∈ B) ∧ ∀x ∈ ∼ C(x ∈ A → x ∈ B)))
16	14, 15	bitri 240	. 2 ⊢ (A ⊆ B ↔ (∀x ∈ C (x ∈ A → x ∈ B) ∧ ∀x ∈ ∼ C(x ∈ A → x ∈ B)))
17	9, 16	syl6rbbr 255	1 ⊢ (A ⊆ C → (A ⊆ B ↔ ∀x ∈ C (x ∈ A → x ∈ B)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∼ ccompl 3206   ∪ cun 3208   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  ssofeq  4078  ssrelk  4212