Theorem ne0i 3264

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2616. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3263	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2325	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1434   =/= wne 2246   (/)c0 3258

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-nul 3259

This theorem is referenced by:  vn0  3265  inelcm  3311  rzal  3346  rexn0  3347  snnzg  3515  prnz  3520  tpnz  3523  onn0  4163  nn0eln0  4367  ordge1n0im  6083  nnmord  6156  phpm  6400  addclpi  6579  mulclpi  6580  uzn0  8715  iccsupr  9065