Theorem elALT 5397

Description: Alternate proof of el 5394, shorter but requiring ax-sep 5243. (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 3446	. 2 ⊢ 𝑥 ∈ V
2		selsALT 5396	. 2 ⊢ (𝑥 ∈ V → ∃𝑦 𝑥 ∈ 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:  ∃wex 1781   ∈ wcel 2114  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908  df-sn 4583  df-pr 4585

This theorem is referenced by: (None)