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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 )