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Metamath Proof Explorer

Theorem opnzi

Description: An ordered pair is nonempty if the arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015)

Ref Expression
Hypotheses opth1.1 ⊢ A ∈ V
opth1.2 ⊢ B ∈ V
Assertion opnzi ⊢ A B ≠ ∅

Proof

Step Hyp Ref Expression
1 opth1.1 ⊢ A ∈ V
2 opth1.2 ⊢ B ∈ V
3 opnz ⊢ A B ≠ ∅ ↔ A ∈ V ∧ B ∈ V
4 1 2 3 mpbir2an ⊢ A B ≠ ∅