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Theorem n0i 3368
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2702. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3367 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2203 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 664 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 616 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1331  wcel 1480  c0 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364
This theorem is referenced by:  ne0i  3369  n0ii  3371  unidif0  4091  iin0r  4093  nnm00  6425  dif1enen  6774  enq0tr  7254
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