Theorem undifr

Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023)

	Ref	Expression
Assertion	undifr	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 )

Proof

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 )