Theorem n0r 3526

Description: An inhabited class is nonempty. See n0rf 3525 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 2386	. 2 |- F/_ x A
2	1	n0rf 3525	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -> wi 4   E.wex 1541   e. wcel 2205   =/= wne 2414   (/)c0 3512

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-dif 3216  df-nul 3513

This theorem is referenced by:  neq0r  3527  opnzi  4356  elqsn0  6851  fin0  7155  infn0  7178  fiubm  11220  lswex  11301  fsumcllem  12110  fprodcllem  12317  setsfun0  13332  gsumwsubmcl  13751  gsumwmhm  13753  g0wlk0  16491