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Theorem elsymdif 4041
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 3946 . . 3 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ (𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)))
2 eldif 3773 . . . 4 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
3 eldif 3773 . . . 4 (𝐴 ∈ (𝐶𝐵) ↔ (𝐴𝐶 ∧ ¬ 𝐴𝐵))
42, 3orbi12i 929 . . 3 ((𝐴 ∈ (𝐵𝐶) ∨ 𝐴 ∈ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
51, 4bitri 266 . 2 (𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
6 df-symdif 4036 . . 3 (𝐵𝐶) = ((𝐵𝐶) ∪ (𝐶𝐵))
76eleq2i 2873 . 2 (𝐴 ∈ (𝐵𝐶) ↔ 𝐴 ∈ ((𝐵𝐶) ∪ (𝐶𝐵)))
8 xor 1029 . 2 (¬ (𝐴𝐵𝐴𝐶) ↔ ((𝐴𝐵 ∧ ¬ 𝐴𝐶) ∨ (𝐴𝐶 ∧ ¬ 𝐴𝐵)))
95, 7, 83bitr4i 294 1 (𝐴 ∈ (𝐵𝐶) ↔ ¬ (𝐴𝐵𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wo 865  wcel 2155  cdif 3760  cun 3761  csymdif 4035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-v 3389  df-dif 3766  df-un 3768  df-symdif 4036
This theorem is referenced by:  dfsymdif4  4042  elsymdifxor  4043  symdifassOLD  4046  brsymdif  4896
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