Theorem xpcoidgend 13714

Description: If two classes are not disjoint, then the composition of their cross-product with itself is idempotent. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
xpcoidgend.1	|- ( ph -> ( A i^i B ) =/= (/) )

Assertion

Ref	Expression
xpcoidgend	|- ( ph -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )

Proof of Theorem xpcoidgend

Step	Hyp	Ref	Expression
1		incom 3805	. . 3 |- ( A i^i B ) = ( B i^i A )
2		xpcoidgend.1	. . 3 |- ( ph -> ( A i^i B ) =/= (/) )
3	1, 2	syl5eqner 2869	. 2 |- ( ph -> ( B i^i A ) =/= (/) )
4	3	xpcogend 13713	1 |- ( ph -> ( ( A X. B ) o. ( A X. B ) ) = ( A X. B ) )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   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-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-co 5123

This theorem is referenced by:  xptrrel  13719  relexpxpnnidm  37995