Theorem n0r 3464

Description: An inhabited class is nonempty. See n0rf 3463 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 2339	. 2 |- F/_ x A
2	1	n0rf 3463	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -> wi 4   E.wex 1506   e. wcel 2167   =/= wne 2367   (/)c0 3450

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  neq0r  3465  opnzi  4268  elqsn0  6663  fin0  6946  infn0  6966  fiubm  10920  fsumcllem  11564  fprodcllem  11771  setsfun0  12714  gsumwsubmcl  13128  gsumwmhm  13130