Theorem ne0i 3415

Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2742. (Contributed by NM, 31-Dec-1993.)

Assertion

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

Proof of Theorem ne0i

Step	Hyp	Ref	Expression
1		n0i 3414	. 2 |- ( B e. A -> -. A = (/) )
2	1	neneqad 2415	1 |- ( B e. A -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2136   =/= wne 2336   (/)c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  ne0d  3416  ne0ii  3418  vn0  3419  inelcm  3469  rzal  3506  rexn0  3507  snnzg  3693  prnz  3698  tpnz  3701  brne0  4031  onn0  4378  nn0eln0  4597  ordge1n0im  6404  nnmord  6485  map0g  6654  phpm  6831  fiintim  6894  addclpi  7268  mulclpi  7269  uzn0  9481  iccsupr  9902