Theorem neq0r 3423

Description: An inhabited class is nonempty. See n0rf 3421 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 3422	. 2 |- ( E. x x e. A -> A =/= (/) )
2	1	neneqd 2357	1 |- ( E. x x e. A -> -. A = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   E.wex 1480   e. wcel 2136   (/)c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  exmidsssn  4181  fzn  9977