Theorem xpidtr 5518

Description: A square Cartesian 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 5147	. . . . . 6 |- ( x ( A X. A ) y <-> ( x e. A /\ y e. A ) )
2		brxp 5147	. . . . . . . . 9 |- ( y ( A X. A ) z <-> ( y e. A /\ z e. A ) )
3		brxp 5147	. . . . . . . . . 10 |- ( x ( A X. A ) z <-> ( x e. A /\ z e. A ) )
4	3	simplbi2com 657	. . . . . . . . 9 |- ( z e. A -> ( x e. A -> x ( A X. A ) z ) )
5	2, 4	simplbiim 659	. . . . . . . 8 |- ( y ( A X. A ) z -> ( x e. A -> x ( A X. A ) z ) )
6	5	com12 32	. . . . . . 7 |- ( x e. A -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
7	6	adantr 481	. . . . . 6 |- ( ( x e. A /\ y e. A ) -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
8	1, 7	sylbi 207	. . . . 5 |- ( x ( A X. A ) y -> ( y ( A X. A ) z -> x ( A X. A ) z ) )
9	8	imp 445	. . . 4 |- ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
10	9	ax-gen 1722	. . 3 |- A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
11	10	gen2 1723	. 2 |- A. x A. y A. z ( ( x ( A X. A ) y /\ y ( A X. A ) z ) -> x ( A X. A ) z )
12		cotr 5508	. 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 ) )
13	11, 12	mpbir 221	1 |- ( ( A X. A ) o. ( A X. A ) ) C_ ( A X. A )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   e. wcel 1990   C_ wss 3574   class class class wbr 4653   X. cxp 5112   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  trinxp  5521  xpider  7818  trust  22033  rtrclex  37924  rtrclexi  37928