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Theorem n0i 3291
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2635. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3290 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2151 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 635 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 592 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1289  wcel 1438  c0 3286
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-nul 3287
This theorem is referenced by:  ne0i  3292  unidif0  4002  iin0r  4004  nnm00  6286  dif1enen  6594  enq0tr  6991
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