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Theorem elsymdif 4263
Description: Membership in a symmetric difference. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
elsymdif (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))

Proof of Theorem elsymdif
StepHypRef Expression
1 elun 4162 . . 3 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ (𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)))
2 eldif 3972 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
3 eldif 3972 . . . 4 (𝐴 ∈ (𝐶𝐵) ↔ (𝐴𝐶 ∧ ¬ 𝐴𝐵))
42, 3orbi12i 914 . . 3 ((𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
51, 4bitri 275 . 2 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
6 df-symdif 4258 . . 3 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
76eleq2i 2830 . 2 (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)))
8 xor 1016 . 2 (¬ (𝐴𝐵𝐴𝐶) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
95, 7, 83bitr4i 303 1 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wo 847  wcel 2105  cdif 3959  cun 3960  csymdif 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-un 3967  df-symdif 4258
This theorem is referenced by:  dfsymdif4  4264  elsymdifxor  4265  brsymdif  5206
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