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Theorem n0i 3206
 Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2547. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i (B A → ¬ A = ∅)

Proof of Theorem n0i
StepHypRef Expression
1 noel 3205 . . 3 ¬ B
2 eleq2 2083 . . 3 (A = ∅ → (B AB ∅))
31, 2mtbiri 587 . 2 (A = ∅ → ¬ B A)
43con2i 545 1 (B A → ¬ A = ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1228   ∈ wcel 1374  ∅c0 3201 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-nul 3202 This theorem is referenced by:  ne0i  3207  unidif0  3894  iin0r  3896  nnm00  6013  enq0tr  6289
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