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Metamath Proof Explorer

Theorem velcomp

Description: Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011)

Ref Expression
Assertion velcomp ⊢ x ∈ V ∖ A ↔ ¬ x ∈ A

Proof

Step Hyp Ref Expression
1 vex ⊢ x ∈ V
2 eldif ⊢ x ∈ V ∖ A ↔ x ∈ V ∧ ¬ x ∈ A
3 1 2 mpbiran ⊢ x ∈ V ∖ A ↔ ¬ x ∈ A