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Metamath Proof Explorer

Theorem elex2

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011) Avoid ax-9 , ax-ext , df-clab . (Revised by Wolf Lammen, 30-Nov-2024)

		Ref	Expression
	Assertion	elex2	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dfclel	⊢ ( 𝐴 ∈ 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) )
2		exsimpr	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵 ) → ∃ 𝑥 𝑥 ∈ 𝐵 )
3	1 2	sylbi	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 ∈ 𝐵 )