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Theorem inteximm 4042
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 3754 . . 3  |-  ( x  e.  A  ->  |^| A  C_  x )
2 vex 2661 . . . 4  |-  x  e. 
_V
32ssex 4033 . . 3  |-  ( |^| A  C_  x  ->  |^| A  e.  _V )
41, 3syl 14 . 2  |-  ( x  e.  A  ->  |^| A  e.  _V )
54exlimiv 1560 1  |-  ( E. x  x  e.  A  ->  |^| A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1451    e. wcel 1463   _Vcvv 2658    C_ wss 3039   |^|cint 3739
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-int 3740
This theorem is referenced by:  intexabim  4045  iinexgm  4047  onintonm  4401  elfi2  6826  elfir  6827  fifo  6834
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