Theorem elsymdif 14727

Description: Membership in a symmetric difference.

Assertion

Ref	Expression
elsymdif	|- (A e. (B ⊕ C) <-> -. (A e. B <-> A e. C))

Proof of Theorem elsymdif

Step	Hyp	Ref	Expression
1		elun 2960	. . 3 |- (A e. ((B \ C) u. (C \ B)) <-> (A e. (B \ C) \/ A e. (C \ B)))
2		eldif 2840	. . . 4 |- (A e. (B \ C) <-> (A e. B /\ -. A e. C))
3		eldif 2840	. . . 4 |- (A e. (C \ B) <-> (A e. C /\ -. A e. B))
4	2, 3	orbi12i 479	. . 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 279	. 2 |- (A e. ((B \ C) u. (C \ B)) <-> ((A e. B /\ -. A e. C) \/ (A e. C /\ -. A e. B)))
6		df-symdif 14722	. . 3 |- (B ⊕ C) = ((B \ C) u. (C \ B))
7	6	eleq2i 2208	. 2 |- (A e. (B ⊕ C) <-> A e. ((B \ C) u. (C \ B)))
8		xor 989	. 2 |- (-. (A e. B <-> A e. C) <-> ((A e. B /\ -. A e. C) \/ (A e. C /\ -. A e. B)))
9	5, 7, 8	3bitr4i 295	1 |- (A e. (B ⊕ C) <-> -. (A e. B <-> A e. C))

Syntax hints:  -. wn 2   <-> wb 219   \/ wo 336   /\ wa 337   e. wcel 1588   \ cdif 2824   u. cun 2825   ⊕ csymdif 14721

This theorem is referenced by:  symdifass 14731  brsymdif 14732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-v 2540  df-dif 2830  df-un 2832  df-symdif 14722

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