ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  inteximm Unicode version

Theorem inteximm 4110
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 3822 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 vex 2715 . . . 4  |-  x  e. 
_V
32ssex 4101 . . 3  |-  ( |^| A  C_  x  ->  |^| A  e.  _V )
41, 3syl 14 . 2  |-  ( x  e.  A  ->  |^| A  e.  _V )
54exlimiv 1578 1  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1472    e. wcel 2128   _Vcvv 2712    C_ wss 3102   |^|cint 3807
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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-int 3808
This theorem is referenced by:  intexabim  4113  iinexgm  4115  onintonm  4476  elfi2  6916  elfir  6917  fifo  6924
  Copyright terms: Public domain W3C validator