Theorem ne0i 3257

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2615. (Contributed by NM, 31-Dec-1993.)

Assertion

Ref	Expression
ne0i	|- ( B e. A -> A =/= (/) )

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3256	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2324	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1433   =/= wne 2245   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by:  vn0  3258  inelcm  3304  rzal  3338  rexn0  3339  snnzg  3507  prnz  3512  tpnz  3515  onn0  4155  nn0eln0  4359  ordge1n0im  6042  nnmord  6113  phpm  6351  addclpi  6517  mulclpi  6518  uzn0  8634  iccsupr  8989