Theorem uneqdifeqim 3613

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 3373	. . . 4 |- ( B u. A ) = ( A u. B )
2		eqtr 2256	. . . . . 6 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> ( B u. A ) = C )
3	2	eqcomd 2244	. . . . 5 |- ( ( ( B u. A ) = ( A u. B ) /\ ( A u. B ) = C ) -> C = ( B u. A ) )
4		difeq1 3340	. . . . . 6 |- ( C = ( B u. A ) -> ( C \ A ) = ( ( B u. A ) \ A ) )
5		difun2 3607	. . . . . 6 |- ( ( B u. A ) \ A ) = ( B \ A )
6		eqtr 2256	. . . . . . 7 |- ( ( ( C \ A ) = ( ( B u. A ) \ A ) /\ ( ( B u. A ) \ A ) = ( B \ A ) ) -> ( C \ A ) = ( B \ A ) )
7		incom 3421	. . . . . . . . . 10 |- ( A i^i B ) = ( B i^i A )
8	7	eqeq1i 2246	. . . . . . . . 9 |- ( ( A i^i B ) = (/) <-> ( B i^i A ) = (/) )
9		disj3 3577	. . . . . . . . 9 |- ( ( B i^i A ) = (/) <-> B = ( B \ A ) )
10	8, 9	bitri 184	. . . . . . . 8 |- ( ( A i^i B ) = (/) <-> B = ( B \ A ) )
11		eqtr 2256	. . . . . . . . . 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 2241	. . . . . . . 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 417	. . . . 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 428	. . 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 1402   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521

This theorem is referenced by:  hashfibclem  11265