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Theorem posprsr 25343
Description: A partial order is a preset. (Contributed by FL, 1-May-2011.)
Assertion
Ref Expression
posprsr  |-  PosetRel  C_ PresetRel

Proof of Theorem posprsr
StepHypRef Expression
1 inss1 3402 . . . . . 6  |-  ( r  i^i  `' r ) 
C_  r
2 sseq1 3212 . . . . . 6  |-  ( (  _I  |`  U. U. r
)  =  ( r  i^i  `' r )  ->  ( (  _I  |`  U. U. r ) 
C_  r  <->  ( r  i^i  `' r )  C_  r ) )
31, 2mpbiri 224 . . . . 5  |-  ( (  _I  |`  U. U. r
)  =  ( r  i^i  `' r )  ->  (  _I  |`  U. U. r )  C_  r
)
43eqcoms 2299 . . . 4  |-  ( ( r  i^i  `' r )  =  (  _I  |`  U. U. r )  ->  (  _I  |`  U. U. r )  C_  r
)
543anim3i 1139 . . 3  |-  ( ( Rel  r  /\  (
r  o.  r ) 
C_  r  /\  (
r  i^i  `' r
)  =  (  _I  |`  U. U. r ) )  ->  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) )
65ss2abi 3258 . 2  |-  { r  |  ( Rel  r  /\  ( r  o.  r
)  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r
) ) }  C_  { r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) }
7 df-ps 14322 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
8 df-prs 25326 . 2  |- PresetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) }
96, 7, 83sstr4i 3230 1  |-  PosetRel  C_ PresetRel
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1632   {cab 2282    i^i cin 3164    C_ wss 3165   U.cuni 3843    _I cid 4320   `'ccnv 4704    |` cres 4707    o. ccom 4709   Rel wrel 4710   PosetRelcps 14317  PresetRelcpresetrel 25318
This theorem is referenced by:  posispre  25344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-ss 3179  df-ps 14322  df-prs 25326
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