Theorem snexOLD 5378

Description: Obsolete version of snex 5375 as of 6-Mar-2026. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snexOLD	⊢ {𝐴} ∈ V

Proof of Theorem snexOLD

Step	Hyp	Ref	Expression
1		snexg 5376	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4656	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5236	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2848	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 183	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1547   ∈ wcel 2119  Vcvv 3432  ∅c0 4268  {csn 4562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-dif 3893  df-un 3895  df-nul 4269  df-sn 4563  df-pr 4565

This theorem is referenced by:  prexOLD  5379