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Theorem xpidtr 4999
Description: A square cross 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 4640 . . . . . 6  |-  ( x ( A  X.  A
) y  <->  ( x  e.  A  /\  y  e.  A ) )
2 brxp 4640 . . . . . . . . 9  |-  ( y ( A  X.  A
) z  <->  ( y  e.  A  /\  z  e.  A ) )
3 brxp 4640 . . . . . . . . . . 11  |-  ( x ( A  X.  A
) z  <->  ( x  e.  A  /\  z  e.  A ) )
43simplbi2com 1437 . . . . . . . . . 10  |-  ( z  e.  A  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
54adantl 275 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( x  e.  A  ->  x ( A  X.  A ) z ) )
62, 5sylbi 120 . . . . . . . 8  |-  ( y ( A  X.  A
) z  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
76com12 30 . . . . . . 7  |-  ( x  e.  A  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
87adantr 274 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
91, 8sylbi 120 . . . . 5  |-  ( x ( A  X.  A
) y  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
109imp 123 . . . 4  |-  ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
1110ax-gen 1442 . . 3  |-  A. z
( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x
( A  X.  A
) z )
1211gen2 1443 . 2  |-  A. x A. y A. z ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
13 cotr 4990 . 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 145 1  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1346    e. wcel 2141    C_ wss 3121   class class class wbr 3987    X. cxp 4607    o. ccom 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-co 4618
This theorem is referenced by:  trinxp  5002  xpider  6582
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