Theorem ssconb 3400

Description: Contraposition law for subsets. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
ssconb	⊢ ((A ⊆ C ∧ B ⊆ C) → (A ⊆ (C ∖ B) ↔ B ⊆ (C ∖ A)))

Proof of Theorem ssconb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . . . . 7 ⊢ (A ⊆ C → (x ∈ A → x ∈ C))
2		ssel 3268	. . . . . . 7 ⊢ (B ⊆ C → (x ∈ B → x ∈ C))
3		pm5.1 830	. . . . . . 7 ⊢ (((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
4	1, 2, 3	syl2an 463	. . . . . 6 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
5		con2b 324	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A))
6	5	a1i 10	. . . . . 6 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A)))
7	4, 6	anbi12d 691	. . . . 5 ⊢ ((A ⊆ C ∧ B ⊆ C) → (((x ∈ A → x ∈ C) ∧ (x ∈ A → ¬ x ∈ B)) ↔ ((x ∈ B → x ∈ C) ∧ (x ∈ B → ¬ x ∈ A))))
8		jcab 833	. . . . 5 ⊢ ((x ∈ A → (x ∈ C ∧ ¬ x ∈ B)) ↔ ((x ∈ A → x ∈ C) ∧ (x ∈ A → ¬ x ∈ B)))
9		jcab 833	. . . . 5 ⊢ ((x ∈ B → (x ∈ C ∧ ¬ x ∈ A)) ↔ ((x ∈ B → x ∈ C) ∧ (x ∈ B → ¬ x ∈ A)))
10	7, 8, 9	3bitr4g 279	. . . 4 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → (x ∈ C ∧ ¬ x ∈ B)) ↔ (x ∈ B → (x ∈ C ∧ ¬ x ∈ A))))
11		eldif 3222	. . . . 5 ⊢ (x ∈ (C ∖ B) ↔ (x ∈ C ∧ ¬ x ∈ B))
12	11	imbi2i 303	. . . 4 ⊢ ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ A → (x ∈ C ∧ ¬ x ∈ B)))
13		eldif 3222	. . . . 5 ⊢ (x ∈ (C ∖ A) ↔ (x ∈ C ∧ ¬ x ∈ A))
14	13	imbi2i 303	. . . 4 ⊢ ((x ∈ B → x ∈ (C ∖ A)) ↔ (x ∈ B → (x ∈ C ∧ ¬ x ∈ A)))
15	10, 12, 14	3bitr4g 279	. . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ B → x ∈ (C ∖ A))))
16	15	albidv 1625	. 2 ⊢ ((A ⊆ C ∧ B ⊆ C) → (∀x(x ∈ A → x ∈ (C ∖ B)) ↔ ∀x(x ∈ B → x ∈ (C ∖ A))))
17		dfss2 3263	. 2 ⊢ (A ⊆ (C ∖ B) ↔ ∀x(x ∈ A → x ∈ (C ∖ B)))
18		dfss2 3263	. 2 ⊢ (B ⊆ (C ∖ A) ↔ ∀x(x ∈ B → x ∈ (C ∖ A)))
19	16, 17, 18	3bitr4g 279	1 ⊢ ((A ⊆ C ∧ B ⊆ C) → (A ⊆ (C ∖ B) ↔ B ⊆ (C ∖ A)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∖ cdif 3207   ⊆ 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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260

This theorem is referenced by:  pssdifcom1  3636  pssdifcom2  3637