Theorem uneqdifeqim 3546

Description: Two ways that A and B can "partition" C (when A and B don't overlap and A is a part of C). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)

Assertion

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

Proof of Theorem uneqdifeqim

Step	Hyp	Ref	Expression
1		uncom 3317	. . . 4 |- ( B u. A ) = ( A u. B )
2		eqtr 2223	. . . . . 6 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( B u. A ) = C )
3	2	eqcomd 2211	. . . . 5 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> C = ( B u. A ) )
4		difeq1 3284	. . . . . 6 |- ( C = ( B u. A ) -> ( C \ A ) = ( ( B u. A ) \ A ) )
5		difun2 3540	. . . . . 6 |- ( ( B u. A ) \ A ) = ( B \ A )
6		eqtr 2223	. . . . . . 7 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( C \ A ) = ( B \ A ) )
7		incom 3365	. . . . . . . . . 10 |- ( A i^i B ) = ( B i^i A )
8	7	eqeq1i 2213	. . . . . . . . 9 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
9		disj3 3513	. . . . . . . . 9 |- ( ( B i^i A ) = (/) <-> B = ( B \ A ) )
10	8, 9	bitri 184	. . . . . . . 8 |- ( ( A i^i B ) = (/) <-> B = ( B \ A ) )
11		eqtr 2223	. . . . . . . . . 10 |- ( ( ( C \ A ) = ( B \ A ) /\ ( B \ A ) = B ) -> ( C \ A ) = B )
12	11	expcom 116	. . . . . . . . 9 |- ( ( B \ A ) = B -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
13	12	eqcoms 2208	. . . . . . . 8 |- ( B = ( B \ A ) -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
14	10, 13	sylbi 121	. . . . . . 7 |- ( ( A i^i B ) = (/) -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
15	6, 14	syl5com 29	. . . . . 6 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
16	4, 5, 15	sylancl 413	. . . . 5 |- ( C = ( B u. A ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
17	3, 16	syl 14	. . . 4 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
18	1, 17	mpan 424	. . 3 |- ( ( A u. B ) = C -> ( ( A i^i B ) = (/) -> ( C \ A ) = B ) )
19	18	com12 30	. 2 |- ( ( A i^i B ) = (/) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )
20	19	adantl 277	1 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   \ cdif 3163   u. cun 3164   i^i cin 3165   C_ wss 3166   (/)c0 3460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461

This theorem is referenced by: (None)