Theorem bj-snex 37484

Description: A singleton is a set. See also snex 5395, snexALT 5339. (Contributed by NM, 7-Aug-1994.) Prove it from ax-bj-sn 37482. (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 37483	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4675	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5256	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2869	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 183	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ∅c0 4285  {csn 4581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-nul 5255  ax-bj-sn 37482

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3907  df-nul 4286  df-sn 4582

This theorem is referenced by:  bj-prex  37489