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Theorem reftr 26263
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 2408 . . . 4  |-  U. A  =  U. A
2 eqid 2408 . . . 4  |-  U. B  =  U. B
31, 2refbas 26254 . . 3  |-  ( A Ref B  ->  U. A  =  U. B )
4 eqid 2408 . . . 4  |-  U. C  =  U. C
52, 4refbas 26254 . . 3  |-  ( B Ref C  ->  U. B  =  U. C )
63, 5sylan9eq 2460 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  U. A  =  U. C )
7 refssex 26255 . . . . . 6  |-  ( ( B Ref C  /\  x  e.  C )  ->  E. y  e.  B  x  C_  y )
87ex 424 . . . . 5  |-  ( B Ref C  ->  (
x  e.  C  ->  E. y  e.  B  x  C_  y ) )
98adantl 453 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. y  e.  B  x  C_  y ) )
10 refssex 26255 . . . . . . 7  |-  ( ( A Ref B  /\  y  e.  B )  ->  E. z  e.  A  y  C_  z )
1110ad2ant2r 728 . . . . . 6  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  y  C_  z )
12 sstr2 3319 . . . . . . . 8  |-  ( x 
C_  y  ->  (
y  C_  z  ->  x 
C_  z ) )
1312reximdv 2781 . . . . . . 7  |-  ( x 
C_  y  ->  ( E. z  e.  A  y  C_  z  ->  E. z  e.  A  x  C_  z
) )
1413ad2antll 710 . . . . . 6  |-  ( ( ( 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 15 . . . . 5  |-  ( ( ( A Ref B  /\  B Ref C )  /\  ( y  e.  B  /\  x  C_  y ) )  ->  E. z  e.  A  x  C_  z )
1615rexlimdvaa 2795 . . . 4  |-  ( ( A Ref B  /\  B Ref C )  -> 
( E. y  e.  B  x  C_  y  ->  E. z  e.  A  x  C_  z ) )
179, 16syld 42 . . 3  |-  ( ( A Ref B  /\  B Ref C )  -> 
( x  e.  C  ->  E. z  e.  A  x  C_  z ) )
1817ralrimiv 2752 . 2  |-  ( ( A Ref B  /\  B Ref C )  ->  A. x  e.  C  E. z  e.  A  x  C_  z )
19 refrel 26252 . . . . 5  |-  Rel  Ref
2019brrelex2i 4882 . . . 4  |-  ( B Ref C  ->  C  e.  _V )
2120adantl 453 . . 3  |-  ( ( A Ref B  /\  B Ref C )  ->  C  e.  _V )
221, 4isref 26253 . . 3  |-  ( C  e.  _V  ->  ( A Ref C  <->  ( U. A  =  U. C  /\  A. x  e.  C  E. z  e.  A  x  C_  z ) ) )
2321, 22syl 16 . 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
) ) )
246, 18, 23mpbir2and 889 1  |-  ( ( A Ref B  /\  B Ref C )  ->  A Ref C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   E.wrex 2671   _Vcvv 2920    C_ wss 3284   U.cuni 3979   class class class wbr 4176   Refcref 26234
This theorem is referenced by:  refssfne  26268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-rel 4848  df-ref 26238
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