Theorem bnj216

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypothesis	bnj216.1	⊢ 𝐵 ∈ V
Assertion	bnj216	⊢ ( 𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		bnj216.1	⊢ 𝐵 ∈ V
2	1	sucid	⊢ 𝐵 ∈ suc 𝐵
3		eleq2	⊢ ( 𝐴 = suc 𝐵 → ( 𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵 ) )
4	2 3	mpbiri	⊢ ( 𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴 )