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Theorem bj-intexr 10966
Description: intexr 3945 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-intexr  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )

Proof of Theorem bj-intexr
StepHypRef Expression
1 bj-vprc 10954 . . 3  |-  -.  _V  e.  _V
2 inteq 3659 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3670 . . . . 5  |-  |^| (/)  =  _V
42, 3syl6eq 2131 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2151 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 633 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2303 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434    =/= wne 2249   _Vcvv 2610   (/)c0 3267   |^|cint 3656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bdn 10875  ax-bdel 10879  ax-bdsep 10942
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2612  df-dif 2984  df-nul 3268  df-int 3657
This theorem is referenced by: (None)
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