MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcoidgend Structured version   Unicode version

Theorem xpcoidgend 12956
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
StepHypRef Expression
1 incom 3631 . . 3  |-  ( A  i^i  B )  =  ( B  i^i  A
)
2 xpcoidgend.1 . . 3  |-  ( ph  ->  ( A  i^i  B
)  =/=  (/) )
31, 2syl5eqner 2704 . 2  |-  ( ph  ->  ( B  i^i  A
)  =/=  (/) )
43xpcogend 12955 1  |-  ( ph  ->  ( ( A  X.  B )  o.  ( A  X.  B ) )  =  ( A  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    =/= wne 2598    i^i cin 3412   (/)c0 3737    X. cxp 4820    o. ccom 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-xp 4828  df-co 4831
This theorem is referenced by:  xptrrel  12961  relexpxpnnidm  35662
  Copyright terms: Public domain W3C validator