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Theorem intexr 3932
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 3916 . . 3  |-  -.  _V  e.  _V
2 inteq 3646 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3657 . . . . 5  |-  |^| (/)  =  _V
42, 3syl6eq 2104 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2122 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 610 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2274 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259    e. wcel 1409    =/= wne 2220   _Vcvv 2574   (/)c0 3252   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2948  df-nul 3253  df-int 3644
This theorem is referenced by: (None)
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