Theorem neq0r 3452

Description: An inhabited class is nonempty. See n0rf 3450 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Assertion

Ref	Expression
neq0r	|- ( E. x x e. A -> -. A = (/) )

Distinct variable group:   x, A

Proof of Theorem neq0r

Step	Hyp	Ref	Expression
1		n0r 3451	. 2 |- ( E. x x e. A -> A =/= (/) )
2	1	neneqd 2381	1 |- ( E. x x e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   E.wex 1503   e. wcel 2160   (/)c0 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-v 2754  df-dif 3146  df-nul 3438

This theorem is referenced by:  exmidsssn  4217  fzn  10060