Theorem pssdifcom2 4440

Description: Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)

Assertion

Ref	Expression
pssdifcom2	⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵)))

Proof of Theorem pssdifcom2

Step	Hyp	Ref	Expression
1		ssconb 4089	. . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
2	1	ancoms 460	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
3		difcom 4438	. . . . 5 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴)
4	3	notbii 320	. . . 4 ⊢ (¬ (𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)
5	4	a1i 11	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (¬ (𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴))
6	2, 5	anbi12d 632	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)))
7		dfpss3 4038	. 2 ⊢ (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ (𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵))
8		dfpss3 4038	. 2 ⊢ (𝐴 ⊊ (𝐶 ∖ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴))
9	6, 7, 8	3bitr4g 314	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∖ cdif 3899   ⊆ wss 3902   ⊊ wpss 3903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3921

This theorem is referenced by:  fin2i2  10180