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Theorem snnzb 4286
 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
StepHypRef Expression
1 snprc 4285 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
2 df-ne 2824 . . . 4 ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
32con2bii 346 . . 3 ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
41, 3bitri 264 . 2 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
54con4bii 310 1 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231  ∅c0 3948  {csn 4210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-nul 3949  df-sn 4211 This theorem is referenced by:  lpvtx  26008  elima4  31803
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