Theorem bj-snex 37712

Description: A singleton is a set. See also snex 5415, snexALT 5359. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37710. (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 37711	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4688	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5275	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2874	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 184	1 ⊢ {𝐴} ∈ V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ∅c0 4289  {csn 4594

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-nul 5274  ax-bj-sn 37710

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-v 3460  df-dif 3911  df-nul 4290  df-sn 4595

This theorem is used by:  bj-prex  37717