Theorem elALT 4837

Description: Alternate proof of el 4773, shorter but requiring more axioms. (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 3176	. . 3 ⊢ 𝑥 ∈ V
2	1	snid 4155	. 2 ⊢ 𝑥 ∈ {𝑥}
3		snex 4835	. . 3 ⊢ {𝑥} ∈ V
4		eleq2 2677	. . 3 ⊢ (𝑦 = {𝑥} → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ {𝑥}))
5	3, 4	spcev 3273	. 2 ⊢ (𝑥 ∈ {𝑥} → ∃𝑦 𝑥 ∈ 𝑦)
6	2, 5	ax-mp 5	1 ⊢ ∃𝑦 𝑥 ∈ 𝑦

Syntax hints:  ∃wex 1695   ∈ wcel 1977  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128

This theorem is referenced by: (None)