Theorem elALT 5381

Description: Alternate proof of el 5377, shorter but requiring ax-sep 5218. (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 3435	. 2 ⊢ 𝑥 ∈ V
2		selsALT 5380	. 2 ⊢ (𝑥 ∈ V → ∃𝑦 𝑥 ∈ 𝑦)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:  ∃wex 1786   ∈ wcel 2119  Vcvv 3431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-un 3888  df-sn 4556  df-pr 4558

This theorem is referenced by: (None)