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Metamath Proof Explorer

Theorem iseri

Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd , which avoids the need to provide a "dummy antecedent" ph if there is no natural one to choose. (Contributed by AV, 30-Apr-2021)

Ref Expression
Hypotheses iseri.1 ⊢ Rel 𝑅
iseri.2 ⊢ ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 )
iseri.3 ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 )
iseri.4 ⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 𝑅 𝑥 )
Assertion iseri 𝑅 Er 𝐴

Proof

Step Hyp Ref Expression
1 iseri.1 ⊢ Rel 𝑅
2 iseri.2 ⊢ ( 𝑥 𝑅 𝑦 → 𝑦 𝑅 𝑥 )
3 iseri.3 ⊢ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 )
4 iseri.4 ⊢ ( 𝑥 ∈ 𝐴 ↔ 𝑥 𝑅 𝑥 )
5 1 a1i ⊢ ( ⊤ → Rel 𝑅 )
6 2 adantl ⊢ ( ( ⊤ ∧ 𝑥 𝑅 𝑦 ) → 𝑦 𝑅 𝑥 )
7 3 adantl ⊢ ( ( ⊤ ∧ ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ) → 𝑥 𝑅 𝑧 )
8 4 a1i ⊢ ( ⊤ → ( 𝑥 ∈ 𝐴 ↔ 𝑥 𝑅 𝑥 ) )
9 5 6 7 8 iserd ⊢ ( ⊤ → 𝑅 Er 𝐴 )
10 9 mptru ⊢ 𝑅 Er 𝐴