MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsymdifxor Structured version   Visualization version   GIF version

Theorem elsymdifxor 4266
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 4264 . 2 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
2 df-xor 1509 . 2 ((𝐴𝐵𝐴𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
31, 2bitr4i 278 1 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wxo 1508  wcel 2106  csymdif 4258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-xor 1509  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-symdif 4259
This theorem is referenced by:  dfsymdif2  4267  symdifass  4268
  Copyright terms: Public domain W3C validator