Theorem xpidtr 4735

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.

Step	Hyp	Ref	Expression
1		brxp 4393	. . . . . 6 |- ( x ( A X. A ) y <-> ( x e. A /\ y e. A ) )
2		brxp 4393	. . . . . . . . 9 |- ( y ( A X. A ) z <-> ( y e. A /\ z e. A ) )
3		brxp 4393	. . . . . . . . . . 11 |- ( x ( A X. A ) z <-> ( x e. A /\ z e. A ) )
4	3	simplbi2com 1373	. . . . . . . . . 10 |- ( z e. A -> ( x e. A -> x ( A X. A ) z ) )
5	4	adantl 271	. . . . . . . . 9 |- ( ( y e. A /\ z e. A ) -> ( x e. A -> x ( A X. A ) z ) )
6	2, 5	sylbi 119	. . . . . . . 8 |- ( y ( A X. A ) z -> ( x e. A -> x ( A X. A ) z ) )
7	6	com12 30	. . . . . . 7 |- ( x e. A -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
8	7	adantr 270	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
9	1, 8	sylbi 119	. . . . 5 |- ( x ( A X. A ) y -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
10	9	imp 122	. . . 4 |- ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
11	10	ax-gen 1378	. . 3 |- A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
12	11	gen2 1379	. 2 |- A. x A. y A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
13		cotr 4726	. 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 ) )
14	12, 13	mpbir 144	1 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   e. wcel 1433   C_ wss 2973   class class class wbr 3785   X. cxp 4361   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-co 4372

This theorem is referenced by:  trinxp  4738  xpiderm  6200