Theorem pssdifcom2 4055

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 3743	. . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
2	1	ancoms 469	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
3		difcom 4053	. . . . 5 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴)
4	3	notbii 310	. . . 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 747	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴)))
7		dfpss3 3693	. 2 ⊢ (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ (𝐵 ⊆ (𝐶 ∖ 𝐴) ∧ ¬ (𝐶 ∖ 𝐴) ⊆ 𝐵))
8		dfpss3 3693	. 2 ⊢ (𝐴 ⊊ (𝐶 ∖ 𝐵) ↔ (𝐴 ⊆ (𝐶 ∖ 𝐵) ∧ ¬ (𝐶 ∖ 𝐵) ⊆ 𝐴))
9	6, 7, 8	3bitr4g 303	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊊ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊊ (𝐶 ∖ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∖ cdif 3571   ⊆ wss 3574   ⊊ wpss 3575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  fin2i2  9140