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Metamath Proof Explorer

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 )