Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023)
Ref | Expression | ||
---|---|---|---|
Assertion | undifr | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undif | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ) | |
2 | uncom | ⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) | |
3 | 2 | eqeq1i | ⊢ ( ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ↔ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 ) |
4 | 1 3 | bitri | ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 ) |