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Theorem elsymdif 3811
 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 3715 . . 3 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ (𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)))
2 eldif 3550 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
3 eldif 3550 . . . 4 (𝐴 ∈ (𝐶𝐵) ↔ (𝐴𝐶 ∧ ¬ 𝐴𝐵))
42, 3orbi12i 542 . . 3 ((𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
51, 4bitri 263 . 2 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
6 df-symdif 3806 . . 3 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
76eleq2i 2680 . 2 (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)))
8 xor 931 . 2 (¬ (𝐴𝐵𝐴𝐶) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
95, 7, 83bitr4i 291 1 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∈ wcel 1977   ∖ cdif 3537   ∪ cun 3538   △ csymdif 3805 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-symdif 3806 This theorem is referenced by:  elsymdifxor  3812  symdifass  3815  brsymdif  4641
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