Theorem xpidtr 4994

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 4635	. . . . . 6 |- ( x ( A X. A ) y <-> ( x e. A /\ y e. A ) )
2		brxp 4635	. . . . . . . . 9 |- ( y ( A X. A ) z <-> ( y e. A /\ z e. A ) )
3		brxp 4635	. . . . . . . . . . 11 |- ( x ( A X. A ) z <-> ( x e. A /\ z e. A ) )
4	3	simplbi2com 1432	. . . . . . . . . 10 |- ( z e. A -> ( x e. A -> x ( A X. A ) z ) )
5	4	adantl 275	. . . . . . . . 9 |- ( ( y e. A /\ z e. A ) -> ( x e. A -> x ( A X. A ) z ) )
6	2, 5	sylbi 120	. . . . . . . 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 274	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
9	1, 8	sylbi 120	. . . . 5 |- ( x ( A X. A ) y -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
10	9	imp 123	. . . 4 |- ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
11	10	ax-gen 1437	. . 3 |- A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
12	11	gen2 1438	. 2 |- A. x A. y A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
13		cotr 4985	. 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 145	1 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   e. wcel 2136   C_ wss 3116   class class class wbr 3982   X. cxp 4602   o. ccom 4608

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-co 4613

This theorem is referenced by:  trinxp  4997  xpider  6572