Theorem symdifxor 3439

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 3175	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
2		eldif 3175	. . . 4 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
3	1, 2	orbi12i 766	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( B \ A ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
4		elun 3314	. . 3 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> ( x e. ( A \ B ) \/ x e. ( B \ A ) ) )
5		excxor 1398	. . . 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 764	. . . 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 2320	1 |- ( ( A \ B ) u. ( B \ A ) ) = { x | ( x e. A \/_ x e. B ) }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 710   = wceq 1373   \/_ wxo 1395   e. wcel 2176   {cab 2191   \ cdif 3163   u. cun 3164

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-xor 1396  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-un 3170

This theorem is referenced by: (None)