Theorem xpidtr 5021

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 4659	. . . . . 6 |- ( x ( A X. A ) y <-> ( x e. A /\ y e. A ) )
2		brxp 4659	. . . . . . . . 9 |- ( y ( A X. A ) z <-> ( y e. A /\ z e. A ) )
3		brxp 4659	. . . . . . . . . . 11 |- ( x ( A X. A ) z <-> ( x e. A /\ z e. A ) )
4	3	simplbi2com 1444	. . . . . . . . . 10 |- ( z e. A -> ( x e. A -> x ( A X. A ) z ) )
5	4	adantl 277	. . . . . . . . 9 |- ( ( y e. A /\ z e. A ) -> ( x e. A -> x ( A X. A ) z ) )
6	2, 5	sylbi 121	. . . . . . . 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 276	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
9	1, 8	sylbi 121	. . . . 5 |- ( x ( A X. A ) y -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
10	9	imp 124	. . . 4 |- ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
11	10	ax-gen 1449	. . 3 |- A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
12	11	gen2 1450	. 2 |- A. x A. y A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
13		cotr 5012	. 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 146	1 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1351   e. wcel 2148   C_ wss 3131   class class class wbr 4005   X. cxp 4626   o. ccom 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-co 4637

This theorem is referenced by:  trinxp  5024  xpider  6608