This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem selsALT

Description: Alternate proof of sels , requiring ax-sep but not using el (which is proved from it as elALT ). (especially when the proof of el is inlined in sels ). (Contributed by NM, 4-Jan-2002) Generalize from the proof of elALT . (Revised by BJ, 3-Apr-2019) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion selsALT ⊢ A ∈ V → ∃ x A ∈ x

Proof

Step Hyp Ref Expression
1 snidg ⊢ A ∈ V → A ∈ A
2 snexg ⊢ A ∈ A → A ∈ V
3 snidg ⊢ A ∈ A → A ∈ A
4 eleq2 ⊢ x = A → A ∈ x ↔ A ∈ A
5 2 3 4 spcedv ⊢ A ∈ A → ∃ x A ∈ x
6 1 5 syl ⊢ A ∈ V → ∃ x A ∈ x