Theorem difuncomp 29369

Description: Express a class difference using unions and class complements. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Assertion

Ref	Expression
difuncomp	|- ( A C_ C -> ( A \ B ) = ( C \ ( ( C \ A ) u. B ) ) )

Proof of Theorem difuncomp

Step	Hyp	Ref	Expression
1		incom 3805	. . . 4 |- ( C i^i A ) = ( A i^i C )
2		sseqin2 3817	. . . . 5 |- ( A C_ C <-> ( C i^i A ) = A )
3	2	biimpi 206	. . . 4 |- ( A C_ C -> ( C i^i A ) = A )
4	1, 3	syl5reqr 2671	. . 3 |- ( A C_ C -> A = ( A i^i C ) )
5	4	difeq1d 3727	. 2 |- ( A C_ C -> ( A \ B ) = ( ( A i^i C ) \ B ) )
6		difundi 3879	. . . 4 |- ( C \ ( ( C \ A ) u. B ) ) = ( ( C \ ( C \ A ) ) i^i ( C \ B ) )
7		dfss4 3858	. . . . . 6 |- ( A C_ C <-> ( C \ ( C \ A ) ) = A )
8	7	biimpi 206	. . . . 5 |- ( A C_ C -> ( C \ ( C \ A ) ) = A )
9	8	ineq1d 3813	. . . 4 |- ( A C_ C -> ( ( C \ ( C \ A ) ) i^i ( C \ B ) ) = ( A i^i ( C \ B ) ) )
10	6, 9	syl5eq 2668	. . 3 |- ( A C_ C -> ( C \ ( ( C \ A ) u. B ) ) = ( A i^i ( C \ B ) ) )
11		indif2 3870	. . 3 |- ( A i^i ( C \ B ) ) = ( ( A i^i C ) \ B )
12	10, 11	syl6eq 2672	. 2 |- ( A C_ C -> ( C \ ( ( C \ A ) u. B ) ) = ( ( A i^i C ) \ B ) )
13	5, 12	eqtr4d 2659	1 |- ( A C_ C -> ( A \ B ) = ( C \ ( ( C \ A ) u. B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   \ cdif 3571   u. cun 3572   i^i cin 3573   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-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588

This theorem is referenced by:  ldgenpisyslem1  30226