Theorem elALT 5440

Description: Alternate proof of el 5437, shorter but requiring ax-sep 5299. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elALT	⊢ ∃𝑦 𝑥 ∈ 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem elALT

Step	Hyp	Ref	Expression
1		vex 3477	. 2 ⊢ 𝑥 ∈ V
2		selsALT 5439	. 2 ⊢ (𝑥 ∈ V → ∃𝑦 𝑥 ∈ 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:  ∃wex 1780   ∈ wcel 2105  Vcvv 3473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631

This theorem is referenced by: (None)