Theorem snexOLD 5393

Description: Obsolete version of snex 5390 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 5391	. 2 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		snprc 4670	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4		0ex 5251	. . 3 ⊢ ∅ ∈ V
5	3, 4	eqeltrdi 2864	. 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ∈ V)
6	1, 5	pm2.61i 183	1 ⊢ {𝐴} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1554   ∈ wcel 2136  Vcvv 3448  ∅c0 4280  {csn 4576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-v 3450  df-dif 3902  df-un 3904  df-nul 4281  df-sn 4577  df-pr 4579

This theorem is referenced by:  prexOLD  5394