Theorem snnzb 4682

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 4681	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		df-ne 2958	. . . 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} ≠ ∅)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  Vcvv 3453  ∅c0 4282  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-nul 4283  df-sn 4588

This theorem is used by:  lpvtx  29526  loop1cycl  30624  elima4  36357