Theorem nelpr

Description: A set A not in a pair is neither element of the pair. (Contributed by Thierry Arnoux, 20-Nov-2023)

	Ref	Expression
Assertion	nelpr	⊢ ( 𝐴 ∈ 𝑉 → ( ¬ 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elprg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) )
2	1	notbid	⊢ ( 𝐴 ∈ 𝑉 → ( ¬ 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) ) )
3		neanior	⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ↔ ¬ ( 𝐴 = 𝐵 ∨ 𝐴 = 𝐶 ) )
4	2 3	bitr4di	⊢ ( 𝐴 ∈ 𝑉 → ( ¬ 𝐴 ∈ { 𝐵 , 𝐶 } ↔ ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ) )