Theorem elex2

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011)

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

Proof

Step	Hyp	Ref	Expression
1		eleq1a	⊢ ( 𝐴 ∈ 𝐵 → ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )
2	1	alrimiv	⊢ ( 𝐴 ∈ 𝐵 → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) )
3		elisset	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
4		exim	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝑥 ∈ 𝐵 ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 𝑥 ∈ 𝐵 ) )
5	2 3 4	sylc	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 ∈ 𝐵 )