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_ C /\ B C_ C ) -> ( B C. ( C \ A ) <-> A C. ( C \ B ) ) )

Proof of Theorem pssdifcom2

Step	Hyp	Ref	Expression
1		ssconb 3743	. . . 4 |- ( ( B C_ C /\ A C_ C ) -> ( B C_ ( C \ A ) <-> A C_ ( C \ B ) ) )
2	1	ancoms 469	. . 3 |- ( ( A C_ C /\ B C_ C ) -> ( B C_ ( C \ A ) <-> A C_ ( C \ B ) ) )
3		difcom 4053	. . . . 5 |- ( ( C \ A ) C_ B <-> ( C \ B ) C_ A )
4	3	notbii 310	. . . 4 |- ( -. ( C \ A ) C_ B <-> -. ( C \ B ) C_ A )
5	4	a1i 11	. . 3 |- ( ( A C_ C /\ B C_ C ) -> ( -. ( C \ A ) C_ B <-> -. ( C \ B ) C_ A ) )
6	2, 5	anbi12d 747	. 2 |- ( ( A C_ C /\ B C_ C ) -> ( ( B C_ ( C \ A ) /\ -. ( C \ A ) C_ B ) <-> ( A C_ ( C \ B ) /\ -. ( C \ B ) C_ A ) ) )
7		dfpss3 3693	. 2 |- ( B C. ( C \ A ) <-> ( B C_ ( C \ A ) /\ -. ( C \ A ) C_ B ) )
8		dfpss3 3693	. 2 |- ( A C. ( C \ B ) <-> ( A C_ ( C \ B ) /\ -. ( C \ B ) C_ A ) )
9	6, 7, 8	3bitr4g 303	1 |- ( ( A C_ C /\ B C_ C ) -> ( B C. ( C \ A ) <-> A C. ( C \ B ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \ cdif 3571   C_ wss 3574   C. 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