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Theorem n0i 3229
 Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2570. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3228 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2101 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 600 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 557 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∈ wcel 1393  ∅c0 3224 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-v 2559  df-dif 2920  df-nul 3225 This theorem is referenced by:  ne0i  3230  unidif0  3920  iin0r  3922  nnm00  6102  enq0tr  6532
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