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Theorem bj-snex 37036
Description: A singleton is a set. See also snex 5436, snexALT 5383. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37034. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-snex {𝐴} ∈ V

Proof of Theorem bj-snex
StepHypRef Expression
1 bj-snexg 37035 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
2 snprc 4717 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 216 . . 3 𝐴 ∈ V → {𝐴} = ∅)
4 0ex 5307 . . 3 ∅ ∈ V
53, 4eqeltrdi 2849 . 2 𝐴 ∈ V → {𝐴} ∈ V)
61, 5pm2.61i 182 1 {𝐴} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626
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 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-nul 5306  ax-bj-sn 37034
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-nul 4334  df-sn 4627
This theorem is referenced by:  bj-prex  37041
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