Metamath Proof Explorer

Theorem elec

Description: Membership in an equivalence class. Theorem 72 of Suppes p. 82. (Contributed by NM, 23-Jul-1995)

Ref Expression
Hypotheses elec.1 ⊢ A ∈ V
elec.2 ⊢ B ∈ V
Assertion elec ⊢ A ∈ B R ↔ B R A

Proof

Step Hyp Ref Expression
1 elec.1 ⊢ A ∈ V
2 elec.2 ⊢ B ∈ V
3 elecg ⊢ A ∈ V ∧ B ∈ V → A ∈ B R ↔ B R A
4 1 2 3 mp2an ⊢ A ∈ B R ↔ B R A