Theorem uneqdifeqim 3536

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 3307	. . . 4 |- ( B u. A ) = ( A u. B )
2		eqtr 2214	. . . . . 6 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( B u. A ) = C )
3	2	eqcomd 2202	. . . . 5 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> C = ( B u. A ) )
4		difeq1 3274	. . . . . 6 |- ( C = ( B u. A ) -> ( C \ A ) = ( ( B u. A ) \ A ) )
5		difun2 3530	. . . . . 6 |- ( ( B u. A ) \ A ) = ( B \ A )
6		eqtr 2214	. . . . . . 7 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( C \ A ) = ( B \ A ) )
7		incom 3355	. . . . . . . . . 10 |- ( A i^i B ) = ( B i^i A )
8	7	eqeq1i 2204	. . . . . . . . 9 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
9		disj3 3503	. . . . . . . . 9 |- ( ( B i^i A ) = (/) <-> B = ( B \ A ) )
10	8, 9	bitri 184	. . . . . . . 8 |- ( ( A i^i B ) = (/) <-> B = ( B \ A ) )
11		eqtr 2214	. . . . . . . . . 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 2199	. . . . . . . 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 1364   \ cdif 3154   u. cun 3155   i^i cin 3156   C_ wss 3157   (/)c0 3450

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451

This theorem is referenced by: (None)