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Theorem elALT 5059
Description: Alternate proof of el 4996, shorter but requiring more axioms. (Contributed by NM, 4-Jan-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elALT 𝑦 𝑥𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem elALT
StepHypRef Expression
1 vex 3343 . . 3 𝑥 ∈ V
21snid 4353 . 2 𝑥 ∈ {𝑥}
3 snex 5057 . . 3 {𝑥} ∈ V
4 eleq2 2828 . . 3 (𝑦 = {𝑥} → (𝑥𝑦𝑥 ∈ {𝑥}))
53, 4spcev 3440 . 2 (𝑥 ∈ {𝑥} → ∃𝑦 𝑥𝑦)
62, 5ax-mp 5 1 𝑦 𝑥𝑦
Colors of variables: wff setvar class
Syntax hints:  wex 1853  wcel 2139  {csn 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324
This theorem is referenced by: (None)
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