Theorem undif5

Description: An equality involving class union and class difference. (Contributed by Thierry Arnoux, 26-Jun-2024)

	Ref	Expression
Assertion	undif5	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		difun2	⊢ ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = ( 𝐴 ∖ 𝐵 )
2		disjdif2	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐴 ∖ 𝐵 ) = 𝐴 )
3	1 2	eqtrid	⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( ( 𝐴 ∪ 𝐵 ) ∖ 𝐵 ) = 𝐴 )