Theorem n0r 3407

Description: An inhabited class is nonempty. See n0rf 3406 for more discussion. (Contributed by Jim Kingdon, 31-Jul-2018.)

Assertion

Ref	Expression
n0r	|- ( E. x x e. A -> A =/= (/) )

Distinct variable group:   x, A

Proof of Theorem n0r

Step	Hyp	Ref	Expression
1		nfcv 2299	. 2 |- F/_ x A
2	1	n0rf 3406	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -> wi 4   E.wex 1472   e. wcel 2128   =/= wne 2327   (/)c0 3394

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-v 2714  df-dif 3104  df-nul 3395

This theorem is referenced by:  neq0r  3408  opnzi  4195  elqsn0  6549  fin0  6830  infn0  6850  fsumcllem  11296  fprodcllem  11503  setsfun0  12237