Theorem bj-snex 37001

Description: A singleton is a set. See also snex 5451, snexALT 5401. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 36999. (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 37000	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4742	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5325	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2852	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 182	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-bj-sn 36999

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-nul 4353  df-sn 4649

This theorem is referenced by:  bj-prex  37006