Theorem ssconb 3743

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 3597	. . . . . . 7 ⊢ (𝐴 ⊆ 𝐶 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶))
2		ssel 3597	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐶 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶))
3		pm5.1 902	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
4	1, 2, 3	syl2an 494	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶)))
5		con2b 349	. . . . . . 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 747	. . . . 5 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))))
8		jcab 907	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
9		jcab 907	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐶) ∧ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
10	7, 8, 9	3bitr4g 303	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))))
11		eldif 3584	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵))
12	11	imbi2i 326	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐵)))
13		eldif 3584	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ∖ 𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴))
14	13	imbi2i 326	. . . 4 ⊢ ((𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)) ↔ (𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴)))
15	10, 12, 14	3bitr4g 303	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
16	15	albidv 1849	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴))))
17		dfss2 3591	. 2 ⊢ (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐶 ∖ 𝐵)))
18		dfss2 3591	. 2 ⊢ (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐶 ∖ 𝐴)))
19	16, 17, 18	3bitr4g 303	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐴 ⊆ (𝐶 ∖ 𝐵) ↔ 𝐵 ⊆ (𝐶 ∖ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   ∈ wcel 1990   ∖ cdif 3571   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  pssdifcom1  4054  pssdifcom2  4055  sbthlem1  8070  sbthlem2  8071  rpnnen2lem11  14953  setscom  15903  dpjidcl  18457  clsval2  20854  regsep2  21180  ordtconnlem1  29970  conss2  38647