Theorem ssconb 4137

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 3975	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		ssel 3975	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
3		pm5.1 821	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
4	1, 2, 3	syl2an 595	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
5		con2b 359	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
6	5	a1i 11	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
7	4, 6	anbi12d 630	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))))
8		jcab 517	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
9		jcab 517	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
10	7, 8, 9	3bitr4g 314	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))))
11		eldif 3958	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))
12	11	imbi2i 336	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))
13		eldif 3958	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
14	13	imbi2i 336	. . . 4 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)))
15	10, 12, 14	3bitr4g 314	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
16	15	albidv 1922	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
17		dfss2 3968	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
18		dfss2 3968	. 2 ⊢ (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)))
19	16, 17, 18	3bitr4g 314	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   ∈ wcel 2105   ∖ cdif 3945   ⊆ wss 3948

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-in 3955  df-ss 3965

This theorem is referenced by:  pssdifcom1  4489  pssdifcom2  4490  sbthlem1  9089  sbthlem2  9090  rpnnen2lem11  16174  setscom  17120  dpjidcl  19976  clsval2  22874  regsep2  23200  cyc3conja  32752  ordtconnlem1  33368  conss2  43665