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Theorem ne0i 3369
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2702. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i  |-  ( B  e.  A  ->  A  =/=  (/) )

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3368 . 2  |-  ( B  e.  A  ->  -.  A  =  (/) )
21neneqad 2387 1  |-  ( B  e.  A  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480    =/= wne 2308   (/)c0 3363
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-nul 3364
This theorem is referenced by:  ne0d  3370  ne0ii  3372  vn0  3373  inelcm  3423  rzal  3460  rexn0  3461  snnzg  3640  prnz  3645  tpnz  3648  brne0  3977  onn0  4322  nn0eln0  4533  ordge1n0im  6333  nnmord  6413  map0g  6582  phpm  6759  fiintim  6817  addclpi  7142  mulclpi  7143  uzn0  9348  iccsupr  9756
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