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Theorem elsymdifxor 4260
Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020.) (Proof shortened by BJ, 13-Aug-2022.)
Assertion
Ref Expression
elsymdifxor (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))

Proof of Theorem elsymdifxor
StepHypRef Expression
1 elsymdif 4258 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
2 df-xor 1512 . 2 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
31, 2bitr4i 278 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wxo 1511  wcel 2108  csymdif 4252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-xor 1512  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-un 3956  df-symdif 4253
This theorem is referenced by:  dfsymdif2  4261  symdifass  4262
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