Theorem xptrrel 13719

Description: The cross product is always a transitive relation. (Contributed by RP, 24-Dec-2019.)

Assertion

Ref	Expression
xptrrel	|- ( ( A X. B ) o. ( A X. B ) ) C_ ( A X. B )

Proof of Theorem xptrrel

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . 8 |- ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ dom ( A X. B )
2		dmxpss 5565	. . . . . . . 8 |- dom ( A X. B ) C_ A
3	1, 2	sstri 3612	. . . . . . 7 |- ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ A
4		inss2 3834	. . . . . . . 8 |- ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ ran ( A X. B )
5		rnxpss 5566	. . . . . . . 8 |- ran ( A X. B ) C_ B
6	4, 5	sstri 3612	. . . . . . 7 |- ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ B
7	3, 6	ssini 3836	. . . . . 6 |- ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ ( A i^i B )
8		eqimss 3657	. . . . . 6 |- ( ( A i^i B ) = (/) -> ( A i^i B ) C_ (/) )
9	7, 8	syl5ss 3614	. . . . 5 |- ( ( A i^i B ) = (/) -> ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ (/) )
10		ss0 3974	. . . . 5 |- ( ( dom ( A X. B ) i^i ran ( A X. B ) ) C_ (/) -> ( dom ( A X. B ) i^i ran ( A X. B ) ) = (/) )
11	9, 10	syl 17	. . . 4 |- ( ( A i^i B ) = (/) -> ( dom ( A X. B ) i^i ran ( A X. B ) ) = (/) )
12	11	coemptyd 13718	. . 3 |- ( ( A i^i B ) = (/) -> ( ( A X. B ) o. ( A X. B ) ) = (/) )
13		0ss 3972	. . 3 |- (/) C_ ( A X. B )
14	12, 13	syl6eqss 3655	. 2 |- ( ( A i^i B ) = (/) -> ( ( A X. B ) o. ( A X. B ) ) C_ ( A X. B ) )
15		df-ne 2795	. . . . 5 |- ( ( A i^i B ) =/= (/) <-> -. ( A i^i B ) = (/) )
16	15	biimpri 218	. . . 4 |- ( -. ( A i^i B ) = (/) -> ( A i^i B ) =/= (/) )
17	16	xpcoidgend 13714	. . 3 |- ( -. ( A i^i B ) = (/) -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )
18		ssid 3624	. . 3 |- ( A X. B ) C_ ( A X. B )
19	17, 18	syl6eqss 3655	. 2 |- ( -. ( A i^i B ) = (/) -> ( ( A X. B ) o. ( A X. B ) ) C_ ( A X. B ) )
20	14, 19	pm2.61i 176	1 |- ( ( A X. B ) o. ( A X. B ) ) C_ ( A X. B )

Syntax hints:   -. wn 3   = wceq 1483   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   X. cxp 5112   dom cdm 5114   ran crn 5115   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-ne 2795  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-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  trclublem  13734  trclubgNEW  37925  trclexi  37927  cnvtrcl0  37933  xpintrreld  37958  trrelsuperreldg  37960  trrelsuperrel2dg  37963