Theorem uneqdifeqim 3443

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 3215	. . . 4 |- ( B u. A ) = ( A u. B )
2		eqtr 2155	. . . . . 6 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( B u. A ) = C )
3	2	eqcomd 2143	. . . . 5 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> C = ( B u. A ) )
4		difeq1 3182	. . . . . 6 |- ( C = ( B u. A ) -> ( C \ A ) = ( ( B u. A ) \ A ) )
5		difun2 3437	. . . . . 6 |- ( ( B u. A ) \ A ) = ( B \ A )
6		eqtr 2155	. . . . . . 7 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( C \ A ) = ( B \ A ) )
7		incom 3263	. . . . . . . . . 10 |- ( A i^i B ) = ( B i^i A )
8	7	eqeq1i 2145	. . . . . . . . 9 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
9		disj3 3410	. . . . . . . . 9 |- ( ( B i^i A ) = (/) <-> B = ( B \ A ) )
10	8, 9	bitri 183	. . . . . . . 8 |- ( ( A i^i B ) = (/) <-> B = ( B \ A ) )
11		eqtr 2155	. . . . . . . . . 10 |- ( ( ( C \ A ) = ( B \ A ) /\ ( B \ A ) = B ) -> ( C \ A ) = B )
12	11	expcom 115	. . . . . . . . 9 |- ( ( B \ A ) = B -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
13	12	eqcoms 2140	. . . . . . . 8 |- ( B = ( B \ A ) -> ( ( C \ A ) = ( B \ A ) -> ( C \ A ) = B ) )
14	10, 13	sylbi 120	. . . . . . 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 409	. . . . 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 420	. . 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 275	1 |- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C -> ( C \ A ) = B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   \ cdif 3063   u. cun 3064   i^i cin 3065   C_ wss 3066   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by: (None)