Theorem bj-snex 37100

Description: A singleton is a set. See also snex 5376, snexALT 5323. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37098. (Revised by BJ, 12-Jan-2025.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-snex	⊢ {𝐴} ∈ V

Proof of Theorem bj-snex

Step	Hyp	Ref	Expression
1		bj-snexg 37099	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4669	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5247	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2841	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 182	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ∅c0 4282  {csn 4575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-nul 5246  ax-bj-sn 37098

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-v 3439  df-dif 3901  df-nul 4283  df-sn 4576

This theorem is referenced by:  bj-prex  37105