Theorem bj-snex 36383

Description: A singleton is a set. See also snex 5431, snexALT 5381. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 36381. (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 36382	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4721	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 215	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5307	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2840	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 182	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2105  Vcvv 3473  ∅c0 4322  {csn 4628

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702  ax-nul 5306  ax-bj-sn 36381

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-nul 4323  df-sn 4629

This theorem is referenced by:  bj-prex  36388