Theorem sscon34b 38317

Description: Relative complementation reverses inclusion of subclasses. Relativized version of complss 3751. (Contributed by RP, 3-Jun-2021.)

Assertion

Ref	Expression
sscon34b	|- ( ( A C_ C /\ B C_ C ) -> ( A C_ B <-> ( C \ B ) C_ ( C \ A ) ) )

Proof of Theorem sscon34b

Step	Hyp	Ref	Expression
1		sscon 3744	. 2 |- ( A C_ B -> ( C \ B ) C_ ( C \ A ) )
2		sscon 3744	. . 3 |- ( ( C \ B ) C_ ( C \ A ) -> ( C \ ( C \ A ) ) C_ ( C \ ( C \ B ) ) )
3		dfss4 3858	. . . . . 6 |- ( A C_ C <-> ( C \ ( C \ A ) ) = A )
4	3	biimpi 206	. . . . 5 |- ( A C_ C -> ( C \ ( C \ A ) ) = A )
5	4	adantr 481	. . . 4 |- ( ( A C_ C /\ B C_ C ) -> ( C \ ( C \ A ) ) = A )
6		dfss4 3858	. . . . . 6 |- ( B C_ C <-> ( C \ ( C \ B ) ) = B )
7	6	biimpi 206	. . . . 5 |- ( B C_ C -> ( C \ ( C \ B ) ) = B )
8	7	adantl 482	. . . 4 |- ( ( A C_ C /\ B C_ C ) -> ( C \ ( C \ B ) ) = B )
9	5, 8	sseq12d 3634	. . 3 |- ( ( A C_ C /\ B C_ C ) -> ( ( C \ ( C \ A ) ) C_ ( C \ ( C \ B ) ) <-> A C_ B ) )
10	2, 9	syl5ib 234	. 2 |- ( ( A C_ C /\ B C_ C ) -> ( ( C \ B ) C_ ( C \ A ) -> A C_ B ) )
11	1, 10	impbid2 216	1 |- ( ( A C_ C /\ B C_ C ) -> ( A C_ B <-> ( C \ B ) C_ ( C \ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   C_ wss 3574

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-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by:  rcompleq  38318  ntrclsss  38361  ntrclsiso  38365  ntrclsk2  38366  ntrclsk3  38368