Theorem elsymdif 3849

Description: Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
elsymdif	|- ( A e. ( B /_\ C ) <-> -. ( A e. B <-> A e. C ) )

Proof of Theorem elsymdif

Step	Hyp	Ref	Expression
1		elun 3753	. . 3 |- ( A e. ( ( B \ C ) u. ( C \ B ) ) <-> ( A e. ( B \ C ) \/ A e. ( C \ B ) ) )
2		eldif 3584	. . . 4 |- ( A e. ( B \ C ) <-> ( A e. B /\ -. A e. C ) )
3		eldif 3584	. . . 4 |- ( A e. ( C \ B ) <-> ( A e. C /\ -. A e. B ) )
4	2, 3	orbi12i 543	. . 3 |- ( ( A e. ( B \ C ) \/ A e. ( C \ B ) ) <-> ( ( A e. B /\ -. A e. C ) \/ ( A e. C /\ -. A e. B ) ) )
5	1, 4	bitri 264	. 2 |- ( A e. ( ( B \ C ) u. ( C \ B ) ) <-> ( ( A e. B /\ -. A e. C ) \/ ( A e. C /\ -. A e. B ) ) )
6		df-symdif 3844	. . 3 |- ( B /_\ C ) = ( ( B \ C ) u. ( C \ B ) )
7	6	eleq2i 2693	. 2 |- ( A e. ( B /_\ C ) <-> A e. ( ( B \ C ) u. ( C \ B ) ) )
8		xor 935	. 2 |- ( -. ( A e. B <-> A e. C ) <-> ( ( A e. B /\ -. A e. C ) \/ ( A e. C /\ -. A e. B ) ) )
9	5, 7, 8	3bitr4i 292	1 |- ( A e. ( B /_\ C ) <-> -. ( A e. B <-> A e. C ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   e. wcel 1990   \ cdif 3571   u. cun 3572   /_\ csymdif 3843

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  df-symdif 3844

This theorem is referenced by:  elsymdifxor  3850  symdifass  3853  brsymdif  4711