Theorem xpidtr 5061

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 4695	. . . . . 6 |- ( x ( A X. A ) y <-> ( x e. A /\ y e. A ) )
2		brxp 4695	. . . . . . . . 9 |- ( y ( A X. A ) z <-> ( y e. A /\ z e. A ) )
3		brxp 4695	. . . . . . . . . . 11 |- ( x ( A X. A ) z <-> ( x e. A /\ z e. A ) )
4	3	simplbi2com 1455	. . . . . . . . . 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 1463	. . 3 |- A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
12	11	gen2 1464	. 2 |- A. x A. y A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
13		cotr 5052	. 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 1362   e. wcel 2167   C_ wss 3157   class class class wbr 4034   X. cxp 4662   o. ccom 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-co 4673

This theorem is referenced by:  trinxp  5064  xpider  6674