Theorem undifr

Description: Union of complementary parts into whole. (Contributed by Thierry Arnoux, 21-Nov-2023) (Proof shortened by SN, 11-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		ssequn2	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐵 ∪ 𝐴 ) = 𝐵 )
2		undif1	⊢ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = ( 𝐵 ∪ 𝐴 )
3	2	eqeq1i	⊢ ( ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 ↔ ( 𝐵 ∪ 𝐴 ) = 𝐵 )
4	1 3	bitr4i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( ( 𝐵 ∖ 𝐴 ) ∪ 𝐴 ) = 𝐵 )