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Theorem bj-snexg 37035
Description: A singleton built on a set is a set. Contrary to bj-snex 37036, this proof is intuitionistically valid and does not require ax-nul 5306. (Contributed by NM, 7-Aug-1994.) Extract it from snex 5436 and prove it from ax-bj-sn 37034. (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 4636 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
2 ax-bj-sn 37034 . . . . 5 𝑥𝑦𝑧(𝑧𝑦𝑧 = 𝑥)
32spi 2184 . . . 4 𝑦𝑧(𝑧𝑦𝑧 = 𝑥)
4 bj-axsn 37033 . . . 4 ({𝑥} ∈ V ↔ ∃𝑦𝑧(𝑧𝑦𝑧 = 𝑥))
53, 4mpbir 231 . . 3 {𝑥} ∈ V
61, 5eqeltrrdi 2850 . 2 (𝑥 = 𝐴 → {𝐴} ∈ V)
76vtocleg 3553 1 (𝐴𝑉 → {𝐴} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  {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-bj-sn 37034
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-sn 4627
This theorem is referenced by:  bj-snex  37036  bj-prexg  37040  bj-adjfrombun  37047
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