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Theorem ne0i 3275
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2626. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (𝐵𝐴𝐴 ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3274 . 2 (𝐵𝐴 → ¬ 𝐴 = ∅)
21neneqad 2328 1 (𝐵𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  wne 2249  c0 3269
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2614  df-dif 2986  df-nul 3270
This theorem is referenced by:  vn0  3276  inelcm  3325  rzal  3360  rexn0  3361  snnzg  3531  prnz  3536  tpnz  3539  onn0  4190  nn0eln0  4395  ordge1n0im  6130  nnmord  6204  map0g  6373  phpm  6509  addclpi  6787  mulclpi  6788  uzn0  8927  iccsupr  9277
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