Theorem uneqdifeqOLD 4058

Description: Obsolete proof of uneqdifeq 4057 as of 14-Jul-2021. (Contributed by FL, 17-Nov-2008.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
uneqdifeqOLD	|- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C <-> ( C \ A ) = B ) )

Proof of Theorem uneqdifeqOLD

Step	Hyp	Ref	Expression
1		uncom 3757	. . . . 5 |- ( B u. A ) = ( A u. B )
2		eqtr 2641	. . . . . . 7 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( B u. A ) = C )
3	2	eqcomd 2628	. . . . . 6 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> C = ( B u. A ) )
4		difeq1 3721	. . . . . . 7 |- ( C = ( B u. A ) -> ( C \ A ) = ( ( B u. A ) \ A ) )
5		difun2 4048	. . . . . . 7 |- ( ( B u. A ) \ A ) = ( B \ A )
6		eqtr 2641	. . . . . . . 8 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( C \ A ) = ( B \ A ) )
7		incom 3805	. . . . . . . . . . 11 |- ( A i^i B ) = ( B i^i A )
8	7	eqeq1i 2627	. . . . . . . . . 10 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
9		disj3 4021	. . . . . . . . . 10 |- ( ( B i^i A ) = (/) <-> B = ( B \ A ) )
10	8, 9	bitri 264	. . . . . . . . 9 |- ( ( A i^i B ) = (/) <-> B = ( B \ A ) )
11		eqtr 2641	. . . . . . . . . . 11 |- ( ( ( C \ A ) = ( B \ A ) /\ ( B \ A ) = B ) -> ( C \ A ) = B )
12	11	expcom 451	. . . . . . . . . 10 |- ( ( B \ A ) = B -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
13	12	eqcoms 2630	. . . . . . . . 9 |- ( B = ( B \ A ) -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
14	10, 13	sylbi 207	. . . . . . . 8 |- ( ( A i^i B ) = (/) -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
15	6, 14	syl5com 31	. . . . . . 7 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
16	4, 5, 15	sylancl 694	. . . . . 6 |- ( C = ( B u. A ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
17	3, 16	syl 17	. . . . 5 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
18	1, 17	mpan 706	. . . 4 |- ( ( A u. B ) = C -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
19	18	com12 32	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )
20	19	adantl 482	. 2 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )
21		difss 3737	. . . . . . . 8 |- ( C \ A ) C_ C
22		sseq1 3626	. . . . . . . . 9 |- ( ( C \ A ) = B -> ( ( C \ A ) C_ C <-> B C_ C ) )
23		unss 3787	. . . . . . . . . . 11 |- ( ( A C_ C /\ B C_ C ) <-> ( A u. B ) C_ C )
24	23	biimpi 206	. . . . . . . . . 10 |- ( ( A C_ C /\ B C_ C ) -> ( A u. B ) C_ C )
25	24	expcom 451	. . . . . . . . 9 |- ( B C_ C -> ( A C_ C -> ( A u. B ) C_ C ) )
26	22, 25	syl6bi 243	. . . . . . . 8 |- ( ( C \ A ) = B -> ( ( C \ A ) C_ C -> ( A C_ C -> ( A u. B ) C_ C ) ) )
27	21, 26	mpi 20	. . . . . . 7 |- ( ( C \ A ) = B -> ( A C_ C -> ( A u. B ) C_ C ) )
28	27	com12 32	. . . . . 6 |- ( A C_ C -> ( ( C \ A ) = B -> ( A u. B ) C_ C ) )
29	28	adantr 481	. . . . 5 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( C \ A ) = B -> ( A u. B ) C_ C ) )
30	29	imp 445	. . . 4 |- ( ( ( A C_ C /\ ( A i^i B ) = (/) ) /\ ( C \ A ) = B ) -> ( A u. B ) C_ C )
31		eqimss 3657	. . . . . . 7 |- ( ( C \ A ) = B -> ( C \ A ) C_ B )
32	31	adantl 482	. . . . . 6 |- ( ( A C_ C /\ ( C \ A ) = B ) -> ( C \ A ) C_ B )
33		ssundif 4052	. . . . . 6 |- ( C C_ ( A u. B ) <-> ( C \ A ) C_ B )
34	32, 33	sylibr 224	. . . . 5 |- ( ( A C_ C /\ ( C \ A ) = B ) -> C C_ ( A u. B ) )
35	34	adantlr 751	. . . 4 |- ( ( ( A C_ C /\ ( A i^i B ) = (/) ) /\ ( C \ A ) = B ) -> C C_ ( A u. B ) )
36	30, 35	eqssd 3620	. . 3 |- ( ( ( A C_ C /\ ( A i^i B ) = (/) ) /\ ( C \ A ) = B ) -> ( A u. B ) = C )
37	36	ex 450	. 2 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( C \ A ) = B -> ( A u. B ) = C ) )
38	20, 37	impbid 202	1 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C <-> ( C \ A ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915

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  df-nul 3916

This theorem is referenced by: (None)