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Metamath Proof Explorer

Theorem sucidg

Description: Part of Proposition 7.23 of TakeutiZaring p. 41 (generalized). Lemma 1.7 of Schloeder p. 1. (Contributed by NM, 25-Mar-1995) (Proof shortened by Scott Fenton, 20-Feb-2012)

Ref Expression
Assertion sucidg ⊢ A ∈ V → A ∈ suc ⁡ A

Proof

Step Hyp Ref Expression
1 eqid ⊢ A = A
2 1 olci ⊢ A ∈ A ∨ A = A
3 elsucg ⊢ A ∈ V → A ∈ suc ⁡ A ↔ A ∈ A ∨ A = A
4 2 3 mpbiri ⊢ A ∈ V → A ∈ suc ⁡ A