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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 )