Theorem snnzb 4723

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 4722	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		df-ne 2939	. . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
3	2	con2bii 357	. . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
4	1, 3	bitri 275	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
5	4	con4bii 321	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ∅c0 4339  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-nul 4340  df-sn 4632

This theorem is referenced by:  lpvtx  29100  loop1cycl  35122  elima4  35757