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Theorem ne0i 3227
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2567. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
ne0i (𝐵𝐴𝐴 ≠ ∅)

Proof of Theorem ne0i
StepHypRef Expression
1 n0i 3226 . 2 (𝐵𝐴 → ¬ 𝐴 = ∅)
21neneqad 2281 1 (𝐵𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1393  wne 2204  c0 3221
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by:  vn0  3228  inelcm  3279  rzal  3315  rexn0  3316  snnzg  3482  prnz  3487  tpnz  3490  onn0  4109  nn0eln0  4304  ordge1n0im  5982  nnmord  6053  phpm  6290  addclpi  6382  mulclpi  6383  uzn0  8436  iccsupr  8778
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