Theorem symdifxor 3430

Description: Expressing symmetric difference with exclusive-or or two differences. (Contributed by Jim Kingdon, 28-Jul-2018.)

Assertion

Ref	Expression
symdifxor	|- ( ( A \ B ) u. ( B \ A ) ) = { x | ( x e. A \/_ x e. B ) }

Distinct variable groups:   x, A   x, B

Proof of Theorem symdifxor

Step	Hyp	Ref	Expression
1		eldif 3166	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
2		eldif 3166	. . . 4 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
3	1, 2	orbi12i 765	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( B \ A ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
4		elun 3305	. . 3 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> ( x e. ( A \ B ) \/ x e. ( B \ A ) ) )
5		excxor 1389	. . . 4 |- ( ( x e. A \/_ x e. B ) <-> ( ( x e. A /\ -. x e. B ) \/ ( -. x e. A /\ x e. B ) ) )
6		ancom 266	. . . . 5 |- ( ( -. x e. A /\ x e. B ) <-> ( x e. B /\ -. x e. A ) )
7	6	orbi2i 763	. . . 4 |- ( ( ( x e. A /\ -. x e. B ) \/ ( -. x e. A /\ x e. B ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
8	5, 7	bitri 184	. . 3 |- ( ( x e. A \/_ x e. B ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
9	3, 4, 8	3bitr4i 212	. 2 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> ( x e. A \/_ x e. B ) )
10	9	abbi2i 2311	1 |- ( ( A \ B ) u. ( B \ A ) ) = { x | ( x e. A \/_ x e. B ) }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 709   = wceq 1364   \/_ wxo 1386   e. wcel 2167   {cab 2182   \ cdif 3154   u. cun 3155

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-xor 1387  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161

This theorem is referenced by: (None)