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Theorem intexr 3904
Description: If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexr  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )

Proof of Theorem intexr
StepHypRef Expression
1 vprc 3888 . . 3  |-  -.  _V  e.  _V
2 inteq 3618 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3629 . . . . 5  |-  |^| (/)  =  _V
42, 3syl6eq 2088 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2106 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 600 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2259 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393    =/= wne 2204   _Vcvv 2557   (/)c0 3224   |^|cint 3615
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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225  df-int 3616
This theorem is referenced by: (None)
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