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 e. _V <-> { A } =/= (/) )

Proof of Theorem snnzb

Step	Hyp	Ref	Expression
1		snprc 4253	. . 3 |- ( -. A e. _V <-> { A } = (/) )
2		df-ne 2795	. . . 4 |- ( { A } =/= (/) <-> -. { A } = (/) )
3	2	con2bii 347	. . 3 |- ( { A } = (/) <-> -. { A } =/= (/) )
4	1, 3	bitri 264	. 2 |- ( -. A e. _V <-> -. { A } =/= (/) )
5	4	con4bii 311	1 |- ( A e. _V <-> { A } =/= (/) )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. 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