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Theorem inteximm 3931
Description: The intersection of an inhabited class exists. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
inteximm  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
Distinct variable group:    x, A

Proof of Theorem inteximm
StepHypRef Expression
1 intss1 3658 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 vex 2577 . . . 4  |-  x  e. 
_V
32ssex 3922 . . 3  |-  ( |^| A  C_  x  ->  |^| A  e.  _V )
41, 3syl 14 . 2  |-  ( x  e.  A  ->  |^| A  e.  _V )
54exlimiv 1505 1  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1397    e. wcel 1409   _Vcvv 2574    C_ wss 2945   |^|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-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-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-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-int 3644
This theorem is referenced by:  intexabim  3934  iinexgm  3936  onintonm  4271
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