Theorem ssconb 3070

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 2933	. . . . . . 7 ⊢ (A ⊆ 𝐶 → (x ∈ A → x ∈ 𝐶))
2		ssel 2933	. . . . . . 7 ⊢ (B ⊆ 𝐶 → (x ∈ B → x ∈ 𝐶))
3		pm5.1 533	. . . . . . 7 ⊢ (((x ∈ A → x ∈ 𝐶) ∧ (x ∈ B → x ∈ 𝐶)) → ((x ∈ A → x ∈ 𝐶) ↔ (x ∈ B → x ∈ 𝐶)))
4	1, 2, 3	syl2an 273	. . . . . 6 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → ((x ∈ A → x ∈ 𝐶) ↔ (x ∈ B → x ∈ 𝐶)))
5		con2b 592	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A))
6	5	a1i 9	. . . . . 6 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A)))
7	4, 6	anbi12d 442	. . . . 5 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → (((x ∈ A → x ∈ 𝐶) ∧ (x ∈ A → ¬ x ∈ B)) ↔ ((x ∈ B → x ∈ 𝐶) ∧ (x ∈ B → ¬ x ∈ A))))
8		jcab 535	. . . . 5 ⊢ ((x ∈ A → (x ∈ 𝐶 ∧ ¬ x ∈ B)) ↔ ((x ∈ A → x ∈ 𝐶) ∧ (x ∈ A → ¬ x ∈ B)))
9		jcab 535	. . . . 5 ⊢ ((x ∈ B → (x ∈ 𝐶 ∧ ¬ x ∈ A)) ↔ ((x ∈ B → x ∈ 𝐶) ∧ (x ∈ B → ¬ x ∈ A)))
10	7, 8, 9	3bitr4g 212	. . . 4 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → ((x ∈ A → (x ∈ 𝐶 ∧ ¬ x ∈ B)) ↔ (x ∈ B → (x ∈ 𝐶 ∧ ¬ x ∈ A))))
11		eldif 2921	. . . . 5 ⊢ (x ∈ (𝐶 ∖ B) ↔ (x ∈ 𝐶 ∧ ¬ x ∈ B))
12	11	imbi2i 215	. . . 4 ⊢ ((x ∈ A → x ∈ (𝐶 ∖ B)) ↔ (x ∈ A → (x ∈ 𝐶 ∧ ¬ x ∈ B)))
13		eldif 2921	. . . . 5 ⊢ (x ∈ (𝐶 ∖ A) ↔ (x ∈ 𝐶 ∧ ¬ x ∈ A))
14	13	imbi2i 215	. . . 4 ⊢ ((x ∈ B → x ∈ (𝐶 ∖ A)) ↔ (x ∈ B → (x ∈ 𝐶 ∧ ¬ x ∈ A)))
15	10, 12, 14	3bitr4g 212	. . 3 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → ((x ∈ A → x ∈ (𝐶 ∖ B)) ↔ (x ∈ B → x ∈ (𝐶 ∖ A))))
16	15	albidv 1702	. 2 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → (∀x(x ∈ A → x ∈ (𝐶 ∖ B)) ↔ ∀x(x ∈ B → x ∈ (𝐶 ∖ A))))
17		dfss2 2928	. 2 ⊢ (A ⊆ (𝐶 ∖ B) ↔ ∀x(x ∈ A → x ∈ (𝐶 ∖ B)))
18		dfss2 2928	. 2 ⊢ (B ⊆ (𝐶 ∖ A) ↔ ∀x(x ∈ B → x ∈ (𝐶 ∖ A)))
19	16, 17, 18	3bitr4g 212	1 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐶) → (A ⊆ (𝐶 ∖ B) ↔ B ⊆ (𝐶 ∖ A)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   ∈ wcel 1390   ∖ cdif 2908   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925

This theorem is referenced by: (None)