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Theorem bj-intexr 11799
Description: intexr 3986 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 11787 . . 3  |-  -.  _V  e.  _V
2 inteq 3691 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3702 . . . . 5  |-  |^| (/)  =  _V
42, 3syl6eq 2136 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2156 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 635 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2309 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438    =/= wne 2255   _Vcvv 2619   (/)c0 3286   |^|cint 3688
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bdn 11708  ax-bdel 11712  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-nul 3287  df-int 3689
This theorem is referenced by: (None)
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