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Theorem relintabex 36232
Description: If the intersection of a class is a relation, then the class is non-empty. (Contributed by RP, 12-Aug-2020.)
Assertion
Ref Expression
relintabex  |-  ( Rel  |^| { x  |  ph }  ->  E. x ph )

Proof of Theorem relintabex
StepHypRef Expression
1 intnex 4574 . . . 4  |-  ( -. 
|^| { x  |  ph }  e.  _V  <->  |^| { x  |  ph }  =  _V )
2 0nelxp 4881 . . . . . . 7  |-  -.  (/)  e.  ( _V  X.  _V )
3 0ex 4549 . . . . . . . 8  |-  (/)  e.  _V
4 eleq1 2528 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
54notbid 300 . . . . . . . 8  |-  ( x  =  (/)  ->  ( -.  x  e.  ( _V 
X.  _V )  <->  -.  (/)  e.  ( _V  X.  _V )
) )
63, 5spcev 3153 . . . . . . 7  |-  ( -.  (/)  e.  ( _V  X.  _V )  ->  E. x  -.  x  e.  ( _V  X.  _V ) )
72, 6ax-mp 5 . . . . . 6  |-  E. x  -.  x  e.  ( _V  X.  _V )
8 nss 3502 . . . . . . . 8  |-  ( -. 
_V  C_  ( _V  X.  _V )  <->  E. x ( x  e.  _V  /\  -.  x  e.  ( _V  X.  _V ) ) )
9 df-rex 2755 . . . . . . . 8  |-  ( E. x  e.  _V  -.  x  e.  ( _V  X.  _V )  <->  E. x
( x  e.  _V  /\ 
-.  x  e.  ( _V  X.  _V )
) )
10 rexv 3074 . . . . . . . 8  |-  ( E. x  e.  _V  -.  x  e.  ( _V  X.  _V )  <->  E. x  -.  x  e.  ( _V  X.  _V ) )
118, 9, 103bitr2i 281 . . . . . . 7  |-  ( -. 
_V  C_  ( _V  X.  _V )  <->  E. x  -.  x  e.  ( _V  X.  _V ) )
12 df-rel 4860 . . . . . . 7  |-  ( Rel 
_V 
<->  _V  C_  ( _V  X.  _V ) )
1311, 12xchnxbir 315 . . . . . 6  |-  ( -. 
Rel  _V  <->  E. x  -.  x  e.  ( _V  X.  _V ) )
147, 13mpbir 214 . . . . 5  |-  -.  Rel  _V
15 releq 4936 . . . . 5  |-  ( |^| { x  |  ph }  =  _V  ->  ( Rel  |^|
{ x  |  ph } 
<->  Rel  _V ) )
1614, 15mtbiri 309 . . . 4  |-  ( |^| { x  |  ph }  =  _V  ->  -.  Rel  |^| { x  |  ph }
)
171, 16sylbi 200 . . 3  |-  ( -. 
|^| { x  |  ph }  e.  _V  ->  -. 
Rel  |^| { x  | 
ph } )
1817con4i 135 . 2  |-  ( Rel  |^| { x  |  ph }  ->  |^| { x  | 
ph }  e.  _V )
19 intexab 4575 . 2  |-  ( E. x ph  <->  |^| { x  |  ph }  e.  _V )
2018, 19sylibr 217 1  |-  ( Rel  |^| { x  |  ph }  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898   {cab 2448   E.wrex 2750   _Vcvv 3057    C_ wss 3416   (/)c0 3743   |^|cint 4248    X. cxp 4851   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-int 4249  df-opab 4476  df-xp 4859  df-rel 4860
This theorem is referenced by:  relintab  36234
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