Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  relsset Structured version   Visualization version   GIF version

Theorem relsset 31979
 Description: The subset class is a relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
relsset Rel SSet

Proof of Theorem relsset
StepHypRef Expression
1 df-sset 31947 . . 3 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2 difss 3735 . . 3 ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ⊆ (V × V)
31, 2eqsstri 3633 . 2 SSet ⊆ (V × V)
4 df-rel 5119 . 2 (Rel SSet SSet ⊆ (V × V))
53, 4mpbir 221 1 Rel SSet
 Colors of variables: wff setvar class Syntax hints:  Vcvv 3198   ∖ cdif 3569   ⊆ wss 3572   E cep 5026   × cxp 5110  ran crn 5113  Rel wrel 5117   ⊗ ctxp 31921   SSet csset 31923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-rel 5119  df-sset 31947 This theorem is referenced by:  brsset  31980  idsset  31981
 Copyright terms: Public domain W3C validator