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Metamath Proof Explorer

Theorem disjr

Description: Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011)

Ref Expression
Assertion disjr ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 ineqcom ⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ( 𝐵 ∩ 𝐴 ) = ∅ )
2 disj ⊢ ( ( 𝐵 ∩ 𝐴 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )
3 1 2 bitri ⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 ∈ 𝐴 )