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Theorem opposet 29980
Description: Every orthoposet is a poset. (Contributed by NM, 12-Oct-2011.)
Assertion
Ref Expression
opposet  |-  ( K  e.  OP  ->  K  e.  Poset )

Proof of Theorem opposet
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2436 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2436 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2436 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2436 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2436 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2436 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29978 . 2  |-  ( K  e.  OP  <->  ( ( K  e.  Poset  /\  ( 0. `  K )  e.  ( Base `  K
)  /\  ( 1. `  K )  e.  (
Base `  K )
)  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) ( ( ( ( oc `  K
) `  x )  e.  ( Base `  K
)  /\  ( ( oc `  K ) `  ( ( oc `  K ) `  x
) )  =  x  /\  ( x ( le `  K ) y  ->  ( ( oc `  K ) `  y ) ( le
`  K ) ( ( oc `  K
) `  x )
) )  /\  (
x ( join `  K
) ( ( oc
`  K ) `  x ) )  =  ( 1. `  K
)  /\  ( x
( meet `  K )
( ( oc `  K ) `  x
) )  =  ( 0. `  K ) ) ) )
9 simpl1 960 . 2  |-  ( ( ( K  e.  Poset  /\  ( 0. `  K
)  e.  ( Base `  K )  /\  ( 1. `  K )  e.  ( Base `  K
) )  /\  A. x  e.  ( Base `  K ) A. y  e.  ( Base `  K
) ( ( ( ( oc `  K
) `  x )  e.  ( Base `  K
)  /\  ( ( oc `  K ) `  ( ( oc `  K ) `  x
) )  =  x  /\  ( x ( le `  K ) y  ->  ( ( oc `  K ) `  y ) ( le
`  K ) ( ( oc `  K
) `  x )
) )  /\  (
x ( join `  K
) ( ( oc
`  K ) `  x ) )  =  ( 1. `  K
)  /\  ( x
( meet `  K )
( ( oc `  K ) `  x
) )  =  ( 0. `  K ) ) )  ->  K  e.  Poset )
108, 9sylbi 188 1  |-  ( K  e.  OP  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   occoc 13537   Posetcpo 14397   joincjn 14401   meetcmee 14402   0.cp0 14466   1.cp1 14467   OPcops 29970
This theorem is referenced by:  op0le  29984  ople0  29985  ople1  29989  op1le  29990  opltcon3b  30002  olposN  30013  ncvr1  30070  cvrcmp2  30082  leatb  30090  dalemcea  30457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-oposet 29974
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