Theorem pssdifcom2 4394

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 4065	. . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
2	1	ancoms 462	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
3		difcom 4392	. . . . 5 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴)
4	3	notbii 323	. . . 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 633	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)))
7		dfpss3 4014	. 2 ⊢ (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ (𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵))
8		dfpss3 4014	. 2 ⊢ (𝐴 ⊊ (𝐶 ∖ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴))
9	6, 7, 8	3bitr4g 317	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∖ cdif 3878   ⊆ wss 3881   ⊊ wpss 3882

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900

This theorem is referenced by:  fin2i2  9729