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Theorem bj-intexr 13908
Description: intexr 4134 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 13896 . . 3  |-  -.  _V  e.  _V
2 inteq 3832 . . . . 5  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3 int0 3843 . . . . 5  |-  |^| (/)  =  _V
42, 3eqtrdi 2219 . . . 4  |-  ( A  =  (/)  ->  |^| A  =  _V )
54eleq1d 2239 . . 3  |-  ( A  =  (/)  ->  ( |^| A  e.  _V  <->  _V  e.  _V ) )
61, 5mtbiri 670 . 2  |-  ( A  =  (/)  ->  -.  |^| A  e.  _V )
76necon2ai 2394 1  |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141    =/= wne 2340   _Vcvv 2730   (/)c0 3414   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-bdn 13817  ax-bdel 13821  ax-bdsep 13884
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415  df-int 3830
This theorem is referenced by: (None)
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