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Theorem n0i 3363
Description: If a set has elements, it is not empty. A set with elements is also inhabited, see elex2 2697. (Contributed by NM, 31-Dec-1993.)
Assertion
Ref Expression
n0i  |-  ( B  e.  A  ->  -.  A  =  (/) )

Proof of Theorem n0i
StepHypRef Expression
1 noel 3362 . . 3  |-  -.  B  e.  (/)
2 eleq2 2201 . . 3  |-  ( A  =  (/)  ->  ( B  e.  A  <->  B  e.  (/) ) )
31, 2mtbiri 664 . 2  |-  ( A  =  (/)  ->  -.  B  e.  A )
43con2i 616 1  |-  ( B  e.  A  ->  -.  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    e. wcel 1480   (/)c0 3358
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 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-nul 3359
This theorem is referenced by:  ne0i  3364  n0ii  3366  unidif0  4086  iin0r  4088  nnm00  6418  dif1enen  6767  enq0tr  7235
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