Theorem bj-snex 38041

Description: A singleton is a set. See also snex 5396, snexALT 5344. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 38039. (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 38040	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4674	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 219	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5260	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2868	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 184	1 ⊢ {𝐴} ∈ V

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  {csn 4584

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-nul 5259  ax-bj-sn 38039

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-nul 4279  df-sn 4585

This theorem is used by:  bj-prex  38046