Theorem cpref 14379

Description: A square cross product (A X. A) is a transitive relation.

Assertion

Ref	Expression
cpref	|- ((A X. A) o. (A X. A)) C_ (A X. A)

Proof of Theorem cpref

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . 7 |- y e. _V
2	1	brxp 4038	. . . . . 6 |- (x(A X. A)y <-> (x e. A /\ y e. A))
3		visset 2295	. . . . . . . . . 10 |- z e. _V
4	3	brxp 4038	. . . . . . . . 9 |- (y(A X. A)z <-> (y e. A /\ z e. A))
5	3	brxp 4038	. . . . . . . . . . . 12 |- (x(A X. A)z <-> (x e. A /\ z e. A))
6	5	biimpri 169	. . . . . . . . . . 11 |- ((x e. A /\ z e. A) -> x(A X. A)z)
7	6	expcom 403	. . . . . . . . . 10 |- (z e. A -> (x e. A -> x(A X. A)z))
8	7	adantl 424	. . . . . . . . 9 |- ((y e. A /\ z e. A) -> (x e. A -> x(A X. A)z))
9	4, 8	sylbi 216	. . . . . . . 8 |- (y(A X. A)z -> (x e. A -> x(A X. A)z))
10	9	com12 14	. . . . . . 7 |- (x e. A -> (y(A X. A)z -> x(A X. A)z))
11	10	adantr 425	. . . . . 6 |- ((x e. A /\ y e. A) -> (y(A X. A)z -> x(A X. A)z))
12	2, 11	sylbi 216	. . . . 5 |- (x(A X. A)y -> (y(A X. A)z -> x(A X. A)z))
13	12	imp 377	. . . 4 |- ((x(A X. A)y /\ y(A X. A)z) -> x(A X. A)z)
14	13	ax-gen 1305	. . 3 |- A.z((x(A X. A)y /\ y(A X. A)z) -> x(A X. A)z)
15	14	gen2 1329	. 2 |- A.xA.yA.z((x(A X. A)y /\ y(A X. A)z) -> x(A X. A)z)
16		cotr 4302	. 2 |- (((A X. A) o. (A X. A)) C_ (A X. A) <-> A.xA.yA.z((x(A X. A)y /\ y(A X. A)z) -> x(A X. A)z))
17	15, 16	mpbir 207	1 |- ((A X. A) o. (A X. A)) C_ (A X. A)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300   C_ wss 2593   class class class wbr 3338   X. cxp 3984   o. ccom 3990

This theorem is referenced by:  sqpeq 14383  tricptr 14385  sqpre 14579

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-co 4003

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