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Theorem bj-snexg 37017
Description: A singleton built on a set is a set. Contrary to bj-snex 37018, this proof is intuitionistically valid and does not require ax-nul 5312. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5442 and prove it from ax-bj-sn 37016. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-snexg (𝐴𝑉 → {𝐴} ∈ V)

Proof of Theorem bj-snexg
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 4641 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
2 ax-bj-sn 37016 . . . . 5 𝑥𝑦𝑧(𝑧𝑦𝑧 = 𝑥)
32spi 2182 . . . 4 𝑦𝑧(𝑧𝑦𝑧 = 𝑥)
4 bj-axsn 37015 . . . 4 ({𝑥} ∈ V ↔ ∃𝑦𝑧(𝑧𝑦𝑧 = 𝑥))
53, 4mpbir 231 . . 3 {𝑥} ∈ V
61, 5eqeltrrdi 2848 . 2 (𝑥 = 𝐴 → {𝐴} ∈ V)
76vtocleg 3553 1 (𝐴𝑉 → {𝐴} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-bj-sn 37016
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sn 4632
This theorem is referenced by:  bj-snex  37018  bj-prexg  37022  bj-adjfrombun  37029
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