Theorem undif

Description: Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998)

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

Proof

Step	Hyp	Ref	Expression
1		ssequn1	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 )
2		undif2	⊢ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = ( 𝐴 ∪ 𝐵 )
3	2	eqeq1i	⊢ ( ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 )
4	1 3	bitr4i	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 )