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Metamath Proof Explorer

Theorem brcnv

Description: The converse of a binary relation swaps arguments. Theorem 11 of Suppes p. 61. (Contributed by NM, 13-Aug-1995)

Ref Expression
Hypotheses opelcnv.1 ⊢ A ∈ V
opelcnv.2 ⊢ B ∈ V
Assertion brcnv ⊢ A R -1 B ↔ B R A

Proof

Step Hyp Ref Expression
1 opelcnv.1 ⊢ A ∈ V
2 opelcnv.2 ⊢ B ∈ V
3 brcnvg ⊢ A ∈ V ∧ B ∈ V → A R -1 B ↔ B R A
4 1 2 3 mp2an ⊢ A R -1 B ↔ B R A