Theorem ssconb 2173

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		pm5.1 678	. . . . . . 7 ⊢ (((x ∈ A → x ∈ C) ⋀ (x ∈ B → x ∈ C)) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
2		ssel 2066	. . . . . . 7 ⊢ (A ⊆ C → (x ∈ A → x ∈ C))
3		ssel 2066	. . . . . . 7 ⊢ (B ⊆ C → (x ∈ B → x ∈ C))
4	1, 2, 3	syl2an 456	. . . . . 6 ⊢ ((A ⊆ C ⋀ B ⊆ C) → ((x ∈ A → x ∈ C) ↔ (x ∈ B → x ∈ C)))
5		bi2.03 165	. . . . . . 7 ⊢ ((x ∈ A → ¬ x ∈ B) ↔ (x ∈ B → ¬ x ∈ A))
6	5	a1i 8	. . . . . 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 600	. . . . 5 ⊢ ((x ∈ A → (x ∈ C ⋀ ¬ x ∈ B)) ↔ ((x ∈ A → x ∈ C) ⋀ (x ∈ A → ¬ x ∈ B)))
9		jcab 600	. . . . 5 ⊢ ((x ∈ B → (x ∈ C ⋀ ¬ x ∈ A)) ↔ ((x ∈ B → x ∈ C) ⋀ (x ∈ B → ¬ x ∈ A)))
10	7, 8, 9	3bitr4g 557	. . . 4 ⊢ ((A ⊆ C ⋀ B ⊆ C) → ((x ∈ A → (x ∈ C ⋀ ¬ x ∈ B)) ↔ (x ∈ B → (x ∈ C ⋀ ¬ x ∈ A))))
11		eldif 2060	. . . . 5 ⊢ (x ∈ (C ∖ B) ↔ (x ∈ C ⋀ ¬ x ∈ B))
12	11	imbi2i 185	. . . 4 ⊢ ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ A → (x ∈ C ⋀ ¬ x ∈ B)))
13		eldif 2060	. . . . 5 ⊢ (x ∈ (C ∖ A) ↔ (x ∈ C ⋀ ¬ x ∈ A))
14	13	imbi2i 185	. . . 4 ⊢ ((x ∈ B → x ∈ (C ∖ A)) ↔ (x ∈ B → (x ∈ C ⋀ ¬ x ∈ A)))
15	10, 12, 14	3bitr4g 557	. . 3 ⊢ ((A ⊆ C ⋀ B ⊆ C) → ((x ∈ A → x ∈ (C ∖ B)) ↔ (x ∈ B → x ∈ (C ∖ A))))
16	15	albidv 1280	. 2 ⊢ ((A ⊆ C ⋀ B ⊆ C) → (∀x(x ∈ A → x ∈ (C ∖ B)) ↔ ∀x(x ∈ B → x ∈ (C ∖ A))))
17		dfss2 2061	. 2 ⊢ (A ⊆ (C ∖ B) ↔ ∀x(x ∈ A → x ∈ (C ∖ B)))
18		dfss2 2061	. 2 ⊢ (B ⊆ (C ∖ A) ↔ ∀x(x ∈ B → x ∈ (C ∖ A)))
19	16, 17, 18	3bitr4g 557	1 ⊢ ((A ⊆ C ⋀ B ⊆ C) → (A ⊆ (C ∖ B) ↔ B ⊆ (C ∖ A)))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   ∈ wcel 960   ∖ cdif 2047   ⊆ wss 2050

This theorem is referenced by:  sbthlem1 4453  sbthlem2 4454  clsval2 7682

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056

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