Theorem snnzb 4680

Description: A singleton is nonempty iff its argument is a set. (Contributed by Scott Fenton, 8-May-2018.)

Assertion

Ref	Expression
snnzb	⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Proof of Theorem snnzb

Step	Hyp	Ref	Expression
1		snprc 4679	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		df-ne 2961	. . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
3	2	con2bii 360	. . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
4	1, 3	bitri 278	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
5	4	con4bii 324	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  Vcvv 3457  ∅c0 4288  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3910  df-nul 4289  df-sn 4586

This theorem is referenced by:  lpvtx  29327  loop1cycl  35500  elima4  36139