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Theorem xpidtr 5387
Description: A square Cartesian product  ( A  X.  A ) is a transitive relation. (Contributed by FL, 31-Jul-2009.)
Assertion
Ref Expression
xpidtr  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)

Proof of Theorem xpidtr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brxp 5029 . . . . . 6  |-  ( x ( A  X.  A
) y  <->  ( x  e.  A  /\  y  e.  A ) )
2 brxp 5029 . . . . . . . . 9  |-  ( y ( A  X.  A
) z  <->  ( y  e.  A  /\  z  e.  A ) )
3 brxp 5029 . . . . . . . . . . 11  |-  ( x ( A  X.  A
) z  <->  ( x  e.  A  /\  z  e.  A ) )
43simplbi2com 627 . . . . . . . . . 10  |-  ( z  e.  A  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
54adantl 466 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( x  e.  A  ->  x ( A  X.  A ) z ) )
62, 5sylbi 195 . . . . . . . 8  |-  ( y ( A  X.  A
) z  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
76com12 31 . . . . . . 7  |-  ( x  e.  A  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
87adantr 465 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
91, 8sylbi 195 . . . . 5  |-  ( x ( A  X.  A
) y  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
109imp 429 . . . 4  |-  ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
1110ax-gen 1601 . . 3  |-  A. z
( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x
( A  X.  A
) z )
1211gen2 1602 . 2  |-  A. x A. y A. z ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
13 cotr 5377 . 2  |-  ( ( ( A  X.  A
)  o.  ( A  X.  A ) ) 
C_  ( A  X.  A )  <->  A. x A. y A. z ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z ) )
1412, 13mpbir 209 1  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377    e. wcel 1767    C_ wss 3476   class class class wbr 4447    X. cxp 4997    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-co 5008
This theorem is referenced by:  trinxp  5390  xpider  7379  trust  20464
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