Theorem ssdifsym

Description: Symmetric class differences for subclasses. (Contributed by AV, 3-Jan-2022)

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

Proof

Step	Hyp	Ref	Expression
1		ssdifim	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 = ( 𝑉 ∖ 𝐴 ) ) → 𝐴 = ( 𝑉 ∖ 𝐵 ) )
2	1	ex	⊢ ( 𝐴 ⊆ 𝑉 → ( 𝐵 = ( 𝑉 ∖ 𝐴 ) → 𝐴 = ( 𝑉 ∖ 𝐵 ) ) )
3	2	adantr	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉 ) → ( 𝐵 = ( 𝑉 ∖ 𝐴 ) → 𝐴 = ( 𝑉 ∖ 𝐵 ) ) )
4		ssdifim	⊢ ( ( 𝐵 ⊆ 𝑉 ∧ 𝐴 = ( 𝑉 ∖ 𝐵 ) ) → 𝐵 = ( 𝑉 ∖ 𝐴 ) )
5	4	ex	⊢ ( 𝐵 ⊆ 𝑉 → ( 𝐴 = ( 𝑉 ∖ 𝐵 ) → 𝐵 = ( 𝑉 ∖ 𝐴 ) ) )
6	5	adantl	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉 ) → ( 𝐴 = ( 𝑉 ∖ 𝐵 ) → 𝐵 = ( 𝑉 ∖ 𝐴 ) ) )
7	3 6	impbid	⊢ ( ( 𝐴 ⊆ 𝑉 ∧ 𝐵 ⊆ 𝑉 ) → ( 𝐵 = ( 𝑉 ∖ 𝐴 ) ↔ 𝐴 = ( 𝑉 ∖ 𝐵 ) ) )