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Theorem n0i 3443
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2768. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (𝐵𝐴 → ¬ 𝐴 = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3441 . . 3 ¬ 𝐵 ∈ ∅
2 eleq2 2253 . . 3 (𝐴 = ∅ → (𝐵𝐴𝐵 ∈ ∅))
31, 2mtbiri 676 . 2 (𝐴 = ∅ → ¬ 𝐵𝐴)
43con2i 628 1 (𝐵𝐴 → ¬ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2160  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by:  ne0i  3444  n0ii  3446  unidif0  4185  iin0r  4187  nnm00  6555  dif1enen  6908  enq0tr  7463
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