Theorem symdif2 3852

Description: Two ways to express symmetric difference. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
symdif2	|- ( ( A \ B ) u. ( B \ A ) ) = { x | -. ( x e. A <-> x e. B ) }

Distinct variable groups:   x, A   x, B

Proof of Theorem symdif2

Step	Hyp	Ref	Expression
1		eldif 3584	. . . 4 |- ( x e. ( A \ B ) <-> ( x e. A /\ -. x e. B ) )
2		eldif 3584	. . . 4 |- ( x e. ( B \ A ) <-> ( x e. B /\ -. x e. A ) )
3	1, 2	orbi12i 543	. . 3 |- ( ( x e. ( A \ B ) \/ x e. ( B \ A ) ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
4		elun 3753	. . 3 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> ( x e. ( A \ B ) \/ x e. ( B \ A ) ) )
5		xor 935	. . 3 |- ( -. ( x e. A <-> x e. B ) <-> ( ( x e. A /\ -. x e. B ) \/ ( x e. B /\ -. x e. A ) ) )
6	3, 4, 5	3bitr4i 292	. 2 |- ( x e. ( ( A \ B ) u. ( B \ A ) ) <-> -. ( x e. A <-> x e. B ) )
7	6	abbi2i 2738	1 |- ( ( A \ B ) u. ( B \ A ) ) = { x | -. ( x e. A <-> x e. B ) }

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   \ cdif 3571   u. cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579

This theorem is referenced by:  mbfeqalem  23409