Theorem symdifxor 3475

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 3210	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
2		eldif 3210	. . . 4 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
3	1, 2	orbi12i 772	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( B \ A ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
4		elun 3350	. . 3 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> ( x e. ( A \ B ) \/ x e. ( B \ A ) ) )
5		excxor 1423	. . . 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 770	. . . 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 2346	1 |- ( ( A \ B ) u. ( B \ A ) ) = { x | ( x e. A \/_ x e. B ) }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 716   = wceq 1398   \/_ wxo 1420   e. wcel 2202   {cab 2217   \ cdif 3198   u. cun 3199

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-xor 1421  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205

This theorem is referenced by: (None)