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Theorem elsymdif 4144
 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 4046 . . 3 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ (𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)))
2 eldif 3869 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
3 eldif 3869 . . . 4 (𝐴 ∈ (𝐶𝐵) ↔ (𝐴𝐶 ∧ ¬ 𝐴𝐵))
42, 3orbi12i 909 . . 3 ((𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
51, 4bitri 276 . 2 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
6 df-symdif 4139 . . 3 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
76eleq2i 2874 . 2 (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)))
8 xor 1009 . 2 (¬ (𝐴𝐵𝐴𝐶) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
95, 7, 83bitr4i 304 1 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∈ wcel 2081   ∖ cdif 3856   ∪ cun 3857   △ csymdif 4138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-dif 3862  df-un 3864  df-symdif 4139 This theorem is referenced by:  dfsymdif4  4145  elsymdifxor  4146  brsymdif  5021
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