Theorem ssdifim

Description: Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022)

	Ref	Expression
Assertion	ssdifim	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 = ( 𝑉 ∖ 𝐴 ) ) → 𝐴 = ( 𝑉 ∖ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		dfss4	⊢ ( 𝐴 ⊆ 𝑉 ↔ ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) = 𝐴 )
2		eqcom	⊢ ( ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) = 𝐴 ↔ 𝐴 = ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) )
3	1 2	sylbb	⊢ ( 𝐴 ⊆ 𝑉 → 𝐴 = ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) )
4		difeq2	⊢ ( 𝐵 = ( 𝑉 ∖ 𝐴 ) → ( 𝑉 ∖ 𝐵 ) = ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) )
5	4	eqcomd	⊢ ( 𝐵 = ( 𝑉 ∖ 𝐴 ) → ( 𝑉 ∖ ( 𝑉 ∖ 𝐴 ) ) = ( 𝑉 ∖ 𝐵 ) )
6	3 5	sylan9eq	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 = ( 𝑉 ∖ 𝐴 ) ) → 𝐴 = ( 𝑉 ∖ 𝐵 ) )