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Theorem posprsr 25240
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 3389 . . . . . 6  |-  ( r  i^i  `' r ) 
C_  r
2 sseq1 3199 . . . . . 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 2286 . . . 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 3245 . 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 14306 . 2  |-  PosetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  ( r  i^i  `' r )  =  (  _I  |`  U. U. r ) ) }
8 df-prs 25223 . 2  |- PresetRel  =  {
r  |  ( Rel  r  /\  ( r  o.  r )  C_  r  /\  (  _I  |`  U. U. r )  C_  r
) }
96, 7, 83sstr4i 3217 1  |-  PosetRel  C_ PresetRel
Colors of variables: wff set class
Syntax hints:    /\ w3a 934    = wceq 1623   {cab 2269    i^i cin 3151    C_ wss 3152   U.cuni 3827    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   Rel wrel 4694   PosetRelcps 14301  PresetRelcpresetrel 25215
This theorem is referenced by:  posispre  25241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-ps 14306  df-prs 25223
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