Theorem ddif

Description: Double complement under universal class. Exercise 4.10(s) of Mendelson p. 231. (Contributed by NM, 8-Jan-2002)

	Ref	Expression
Assertion	ddif	⊢ ( V ∖ ( V ∖ 𝐴 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		velcomp	⊢ ( 𝑥 ∈ ( V ∖ 𝐴 ) ↔ ¬ 𝑥 ∈ 𝐴 )
2	1	con2bii	⊢ ( 𝑥 ∈ 𝐴 ↔ ¬ 𝑥 ∈ ( V ∖ 𝐴 ) )
3		vex	⊢ 𝑥 ∈ V
4	3	biantrur	⊢ ( ¬ 𝑥 ∈ ( V ∖ 𝐴 ) ↔ ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ( V ∖ 𝐴 ) ) )
5	2 4	bitr2i	⊢ ( ( 𝑥 ∈ V ∧ ¬ 𝑥 ∈ ( V ∖ 𝐴 ) ) ↔ 𝑥 ∈ 𝐴 )
6	5	difeqri	⊢ ( V ∖ ( V ∖ 𝐴 ) ) = 𝐴