Theorem snnzb 4685

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 4684	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		df-ne 2927	. . . 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 1540   ∈ wcel 2109   ≠ wne 2926  Vcvv 3450  ∅c0 4299  {csn 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-v 3452  df-dif 3920  df-nul 4300  df-sn 4593

This theorem is referenced by:  lpvtx  29002  loop1cycl  35131  elima4  35770