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Theorem xpidtr 5210
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 4854 . . . . . 6  |-  ( x ( A  X.  A
) y  <->  ( x  e.  A  /\  y  e.  A ) )
2 brxp 4854 . . . . . . . . 9  |-  ( y ( A  X.  A
) z  <->  ( y  e.  A  /\  z  e.  A ) )
3 brxp 4854 . . . . . . . . . . 11  |-  ( x ( A  X.  A
) z  <->  ( x  e.  A  /\  z  e.  A ) )
43simplbi2com 625 . . . . . . . . . 10  |-  ( z  e.  A  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
54adantl 464 . . . . . . . . 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 29 . . . . . . 7  |-  ( x  e.  A  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
87adantr 463 . . . . . 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 427 . . . 4  |-  ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
1110ax-gen 1639 . . 3  |-  A. z
( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x
( A  X.  A
) z )
1211gen2 1640 . 2  |-  A. x A. y A. z ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
13 cotr 5200 . 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 367   A.wal 1403    e. wcel 1842    C_ wss 3414   class class class wbr 4395    X. cxp 4821    o. ccom 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-co 4832
This theorem is referenced by:  trinxp  5213  xpider  7419  trust  21024
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