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Theorem ne0i 3206
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2546. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (B AA ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3205 . 2 (B A → ¬ A = ∅)
21neneqad 2261 1 (B AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  wne 2187  c0 3200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ne 2189  df-v 2536  df-dif 2896  df-nul 3201
This theorem is referenced by:  vn0  3207  inelcm  3258  rzal  3296  rexn0  3297  snnzg  3458  prnz  3463  tpnz  3466  onn0  4084  nn0eln0  4266  ordge1n0im  5929  nnmord  5996  addclpi  6179  mulclpi  6180
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