Theorem n0r 3437

Description: An inhabited class is nonempty. See n0rf 3436 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 2319	. 2 |- F/_ x A
2	1	n0rf 3436	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -> wi 4   E.wex 1492   e. wcel 2148   =/= wne 2347   (/)c0 3423

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-v 2740  df-dif 3132  df-nul 3424

This theorem is referenced by:  neq0r  3438  opnzi  4236  elqsn0  6604  fin0  6885  infn0  6905  fiubm  10808  fsumcllem  11407  fprodcllem  11614  setsfun0  12498