ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpidtr Unicode version

Theorem xpidtr 4715
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 4375 . . . . . 6  |-  ( x ( A  X.  A
) y  <->  ( x  e.  A  /\  y  e.  A ) )
2 brxp 4375 . . . . . . . . 9  |-  ( y ( A  X.  A
) z  <->  ( y  e.  A  /\  z  e.  A ) )
3 brxp 4375 . . . . . . . . . . 11  |-  ( x ( A  X.  A
) z  <->  ( x  e.  A  /\  z  e.  A ) )
43simplbi2com 1333 . . . . . . . . . 10  |-  ( z  e.  A  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
54adantl 262 . . . . . . . . 9  |-  ( ( y  e.  A  /\  z  e.  A )  ->  ( x  e.  A  ->  x ( A  X.  A ) z ) )
62, 5sylbi 114 . . . . . . . 8  |-  ( y ( A  X.  A
) z  ->  (
x  e.  A  ->  x ( A  X.  A ) z ) )
76com12 27 . . . . . . 7  |-  ( x  e.  A  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
87adantr 261 . . . . . 6  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
91, 8sylbi 114 . . . . 5  |-  ( x ( A  X.  A
) y  ->  (
y ( A  X.  A ) z  ->  x ( A  X.  A ) z ) )
109imp 115 . . . 4  |-  ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
1110ax-gen 1338 . . 3  |-  A. z
( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x
( A  X.  A
) z )
1211gen2 1339 . 2  |-  A. x A. y A. z ( ( x ( A  X.  A ) y  /\  y ( A  X.  A ) z )  ->  x ( A  X.  A ) z )
13 cotr 4706 . 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 134 1  |-  ( ( A  X.  A )  o.  ( A  X.  A ) )  C_  ( A  X.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   A.wal 1241    e. wcel 1393    C_ wss 2917   class class class wbr 3764    X. cxp 4343    o. ccom 4349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-co 4354
This theorem is referenced by:  trinxp  4718  xpiderm  6177
  Copyright terms: Public domain W3C validator