Theorem ssconb 4142

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 3977	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		ssel 3977	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
3		pm5.1 824	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
4	1, 2, 3	syl2an 596	. . . . . 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 632	. . . . 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 3961	. . . . 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 3961	. . . . 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 1920	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
17		df-ss 3968	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
18		df-ss 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 206   ∧ wa 395  ∀wal 1538   ∈ wcel 2108   ∖ cdif 3948   ⊆ wss 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968

This theorem is referenced by:  pssdifcom1  4490  pssdifcom2  4491  sbthlem1  9123  sbthlem2  9124  rpnnen2lem11  16260  setscom  17217  dpjidcl  20078  clsval2  23058  regsep2  23384  cyc3conja  33177  ordtconnlem1  33923  conss2  44462