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Theorem intexr 4007
Description: If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexr  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )

Proof of Theorem intexr
StepHypRef Expression
1 vprc 3992 . . 3  |-  -.  _V  e.  _V
2 inteq 3713 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3724 . . . . 5  |-  |^| (/)  =  _V
42, 3syl6eq 2143 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2163 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 638 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2316 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445    =/= wne 2262   _Vcvv 2633   (/)c0 3302   |^|cint 3710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-v 2635  df-dif 3015  df-nul 3303  df-int 3711
This theorem is referenced by: (None)
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