Theorem difdif

Description: Double class difference. Exercise 11 of TakeutiZaring p. 22. (Contributed by NM, 17-May-1998)

	Ref	Expression
Assertion	difdif	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴

Proof

Step	Hyp	Ref	Expression
1		pm4.45im	⊢ ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ) )
2		iman	⊢ ( ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ↔ ¬ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) )
3		eldif	⊢ ( 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴 ) )
4	2 3	xchbinxr	⊢ ( ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ↔ ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) )
5	4	anbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ( 𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) )
6	1 5	bitr2i	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) ) ↔ 𝑥 ∈ 𝐴 )
7	6	difeqri	⊢ ( 𝐴 ∖ ( 𝐵 ∖ 𝐴 ) ) = 𝐴