Theorem snnzb 4254

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 4253	. . 3 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
2		df-ne 2795	. . . 4 ⊢ ({𝐴} ≠ ∅ ↔ ¬ {𝐴} = ∅)
3	2	con2bii 347	. . 3 ⊢ ({𝐴} = ∅ ↔ ¬ {𝐴} ≠ ∅)
4	1, 3	bitri 264	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ {𝐴} ≠ ∅)
5	4	con4bii 311	1 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200  ∅c0 3915  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916  df-sn 4178

This theorem is referenced by:  lpvtx  25963  elima4  31679