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Metamath Proof Explorer

Theorem ecelqsi

Description: Membership of an equivalence class in a quotient set. (Contributed by NM, 25-Jul-1995) (Revised by Mario Carneiro, 9-Jul-2014)

Ref Expression
Hypothesis ecelqsi.1 ⊢ 𝑅 ∈ V
Assertion ecelqsi ( 𝐵 ∈ 𝐴 → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )

Proof

Step Hyp Ref Expression
1 ecelqsi.1 ⊢ 𝑅 ∈ V
2 ecelqsw ⊢ ( ( 𝑅 ∈ V ∧ 𝐵 ∈ 𝐴 ) → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )
3 1 2 mpan ⊢ ( 𝐵 ∈ 𝐴 → [ 𝐵 ] 𝑅 ∈ ( 𝐴 / 𝑅 ) )