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Theorem bj-snex 35438
Description: A singleton is a set. See also snex 5387, snexALT 5337. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 35436. (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 35437 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
2 snprc 4677 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
4 0ex 5263 . . 3 ∅ ∈ V
53, 4eqeltrdi 2847 . 2 𝐴 ∈ V → {𝐴} ∈ V)
61, 5pm2.61i 182 1 {𝐴} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3444  c0 4281  {csn 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2709  ax-nul 5262  ax-bj-sn 35436
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3446  df-dif 3912  df-nul 4282  df-sn 4586
This theorem is referenced by:  bj-prex  35443
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