Theorem pssdifcom1 4054

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

Assertion

Ref	Expression
pssdifcom1	⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐶 ∖ 𝐴) ⊊ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊊ 𝐴))

Proof of Theorem pssdifcom1

Step	Hyp	Ref	Expression
1		difcom 4053	. . . 4 ⊢ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴)
2	1	a1i 11	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐶 ∖ 𝐴) ⊆ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊆ 𝐴))
3		ssconb 3743	. . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐴 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
4	3	ancoms 469	. . . 4 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
5	4	notbid 308	. . 3 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (¬ 𝐵 ⊆ (𝐶 ∖ 𝐴) ↔ ¬ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
6	2, 5	anbi12d 747	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → (((𝐶 ∖ 𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶 ∖ 𝐴)) ↔ ((𝐶 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶 ∖ 𝐵))))
7		dfpss3 3693	. 2 ⊢ ((𝐶 ∖ 𝐴) ⊊ 𝐵 ↔ ((𝐶 ∖ 𝐴) ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ (𝐶 ∖ 𝐴)))
8		dfpss3 3693	. 2 ⊢ ((𝐶 ∖ 𝐵) ⊊ 𝐴 ↔ ((𝐶 ∖ 𝐵) ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ (𝐶 ∖ 𝐵)))
9	6, 7, 8	3bitr4g 303	1 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶) → ((𝐶 ∖ 𝐴) ⊊ 𝐵 ↔ (𝐶 ∖ 𝐵) ⊊ 𝐴))

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:  isfin2-2  9141  compssiso  9196