Theorem n0r 3535

Description: An inhabited class is nonempty. See n0rf 3534 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 2392	. 2 |- F/_ x A
2	1	n0rf 3534	1 |- ( E. x x e. A -> A =/= (/) )

This proof depends on syntax axioms:   -> wi 4   E.wex 1545   e. wcel 2209   =/= wne 2420   (/)c0 3520

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-dif 3222  df-nul 3521

This theorem is used by:  neq0r  3536  opnzi  4375  elqsn0  6878  fin0  7189  infn0  7212  fiubm  11271  lswex  11356  fsumcllem  12166  fprodcllem  12373  setsfun0  13388  gzsumwsubmcl  13801  gzsumwmhm  13803  g0wlk0  16611