Theorem ssconb 1598

Description: Contraposition law for subsets.

Assertion

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

Proof of Theorem ssconb

Step	Hyp	Ref	Expression
1		pm5.1 501	. . . . . . 7 ⊢ (((x ∈ A → x ∈ C) ∧ (x ∈ B → x ∈ C)) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
2		ssel 1502	. . . . . . 7 ⊢ (A ⊆ C → (x ∈ A → x ∈ C))
3		ssel 1502	. . . . . . 7 ⊢ (B ⊆ C → (x ∈ B → x ∈ C))
4	1, 2, 3	syl2an 349	. . . . . 6 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
5		bi2.03 144	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A))
6	5	a1i 7	. . . . . 6 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A)))
7	4, 6	anbi12d 476	. . . . 5 ⊢ ((A ⊆ C ∧ B ⊆ C) → (((x ∈ A → x ∈ C) ∧ (x ∈ A → ¬ x ∈ B)) ↔ ((x ∈ B → x ∈ C) ∧ (x ∈ B → ¬ x ∈ A))))
8		jcab 454	. . . . 5 ⊢ ((x ∈ A → (x ∈ C ∧ ¬ x ∈ B)) ↔ ((x ∈ A → x ∈ C) ∧ (x ∈ A → ¬ x ∈ B)))
9		jcab 454	. . . . 5 ⊢ ((x ∈ B → (x ∈ C ∧ ¬ x ∈ A)) ↔ ((x ∈ B → x ∈ C) ∧ (x ∈ B → ¬ x ∈ A)))
10	7, 8, 9	3bitr4g 428	. . . 4 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → (x ∈ C ∧ ¬ x ∈ B)) ↔ (x ∈ B → (x ∈ C ∧ ¬ x ∈ A))))
11		eldif 1496	. . . . 5 ⊢ (x ∈ (C ∖ B) ↔ (x ∈ C ∧ ¬ x ∈ B))
12	11	imbi2i 160	. . . 4 ⊢ ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ A → (x ∈ C ∧ ¬ x ∈ B)))
13		eldif 1496	. . . . 5 ⊢ (x ∈ (C ∖ A) ↔ (x ∈ C ∧ ¬ x ∈ A))
14	13	imbi2i 160	. . . 4 ⊢ ((x ∈ B → x ∈ (C ∖ A)) ↔ (x ∈ B → (x ∈ C ∧ ¬ x ∈ A)))
15	10, 12, 14	3bitr4g 428	. . 3 ⊢ ((A ⊆ C ∧ B ⊆ C) → ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ B → x ∈ (C ∖ A))))
16	15	bialdv 935	. 2 ⊢ ((A ⊆ C ∧ B ⊆ C) → (∀x(x ∈ A → x ∈ (C ∖ B)) ↔ ∀x(x ∈ B → x ∈ (C ∖ A))))
17		dfss2 1497	. 2 ⊢ (A ⊆ (C ∖ B) ↔ ∀x(x ∈ A → x ∈ (C ∖ B)))
18		dfss2 1497	. 2 ⊢ (B ⊆ (C ∖ A) ↔ ∀x(x ∈ B → x ∈ (C ∖ A)))
19	16, 17, 18	3bitr4g 428	1 ⊢ ((A ⊆ C ∧ B ⊆ C) → (A ⊆ (C ∖ B) ↔ B ⊆ (C ∖ A)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wcel 1092   ∖ cdif 1484   ⊆ wss 1487

This theorem is referenced by:  sbthlem1 3349  sbthlem2 3350

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492

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