Theorem elsymdifxor 4213

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 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

Proof of Theorem elsymdifxor

Step	Hyp	Ref	Expression
1		elsymdif 4211	. 2 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
2		df-xor 1542	. 2 ⊢ ((𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶) ↔ ¬ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶))
3	1, 2	bitr4i 281	1 ⊢ (𝐴 ∈ (𝐵 △ 𝐶) ↔ (𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ⊻ wxo 1541   ∈ wcel 2146   △ csymdif 4205

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-xor 1542  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dif 3909  df-un 3911  df-symdif 4206

This theorem is used by:  dfsymdif2  4214  symdifass  4215