Theorem bj-snex 37017

Description: A singleton is a set. See also snex 5441, snexALT 5388. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37015. (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 37016	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4721	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5312	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2846	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 182	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2105  Vcvv 3477  ∅c0 4338  {csn 4630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-12 2174  ax-ext 2705  ax-nul 5311  ax-bj-sn 37015

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965  df-nul 4339  df-sn 4631

This theorem is referenced by:  bj-prex  37022