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Theorem ne0i 3205
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2545. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (B AA ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3204 . 2 (B A → ¬ A = ∅)
21neneqad 2260 1 (B AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1374  wne 2186  c0 3199
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2535  df-dif 2895  df-nul 3200
This theorem is referenced by:  vn0  3206  inelcm  3257  rzal  3295  rexn0  3296  snnzg  3457  prnz  3462  tpnz  3465  onn0  4084  nn0eln0  4266  ordge1n0im  5932  nnmord  5999  addclpi  6185  mulclpi  6186
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