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Theorem reftr 26289
Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.)
Assertion
Ref Expression
reftr  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )

Proof of Theorem reftr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . 4  |-  U. A  =  U. A
2 eqid 2283 . . . 4  |-  U. B  =  U. B
31, 2refbas 26280 . . 3  |-  ( A Ref B  ->  U. A  =  U. B )
4 eqid 2283 . . . 4  |-  U. C  =  U. C
52, 4refbas 26280 . . 3  |-  ( B Ref C  ->  U. B  =  U. C )
63, 5sylan9eq 2335 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  U. A  =  U. C )
7 refssex 26281 . . . . . 6  |-  ( ( B Ref C  /\  x  e.  C )  ->  E. y  e.  B  x  C_  y )
87ex 423 . . . . 5  |-  ( B Ref C  ->  (
x  e.  C  ->  E. y  e.  B  x  C_  y ) )
98adantl 452 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. y  e.  B  x  C_  y ) )
10 refssex 26281 . . . . . . . 8  |-  ( ( A Ref B  /\  y  e.  B )  ->  E. z  e.  A  y  C_  z )
1110ad2ant2r 727 . . . . . . 7  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  y  C_  z )
12 sstr2 3186 . . . . . . . . 9  |-  ( x 
C_  y  ->  (
y  C_  z  ->  x 
C_  z ) )
1312reximdv 2654 . . . . . . . 8  |-  ( x 
C_  y  ->  ( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z
) )
1413ad2antll 709 . . . . . . 7  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  -> 
( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z ) )
1511, 14mpd 14 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  x  C_  z )
1615exp32 588 . . . . 5  |-  ( ( A Ref B  /\  B Ref C )  -> 
( y  e.  B  ->  ( x  C_  y  ->  E. z  e.  A  x  C_  z ) ) )
1716rexlimdv 2666 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( E. y  e.  B  x  C_  y  ->  E. z  e.  A  x  C_  z ) )
189, 17syld 40 . . 3  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. z  e.  A  x  C_  z ) )
1918ralrimiv 2625 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  A. x  e.  C  E. z  e.  A  x  C_  z )
20 refrel 26278 . . . . 5  |-  Rel  Ref
2120brrelex2i 4730 . . . 4  |-  ( B Ref C  ->  C  e.  _V )
2221adantl 452 . . 3  |-  ( ( A Ref B  /\  B Ref C )  ->  C  e.  _V )
231, 4isref 26279 . . 3  |-  ( C  e.  _V  ->  ( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z ) ) )
2422, 23syl 15 . 2  |-  ( ( A Ref B  /\  B Ref C )  -> 
( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z
) ) )
256, 19, 24mpbir2and 888 1  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   U.cuni 3827   class class class wbr 4023   Refcref 26260
This theorem is referenced by:  refssfne  26294
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-ref 26264
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