Theorem elALT 5400

Description: Alternate proof of el 5397, shorter but requiring ax-sep 5251. (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 3451	. 2 ⊢ 𝑥 ∈ V
2		selsALT 5399	. 2 ⊢ (𝑥 ∈ V → ∃𝑦 𝑥 ∈ 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:  ∃wex 1779   ∈ wcel 2109  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3449  df-un 3919  df-sn 4590  df-pr 4592

This theorem is referenced by: (None)