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 ) ) ) |
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 ) ) ) |