Theorem elALT

Description: Alternate proof of el , shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	elALT	⊢ ∃ 𝑦 𝑥 ∈ 𝑦

Proof

Step	Hyp	Ref	Expression
1		vex	⊢ 𝑥 ∈ V
2	1	snid	⊢ 𝑥 ∈ { 𝑥 }
3		snex	⊢ { 𝑥 } ∈ V
4		eleq2	⊢ ( 𝑦 = { 𝑥 } → ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ { 𝑥 } ) )
5	3 4	spcev	⊢ ( 𝑥 ∈ { 𝑥 } → ∃ 𝑦 𝑥 ∈ 𝑦 )
6	2 5	ax-mp	⊢ ∃ 𝑦 𝑥 ∈ 𝑦