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Theorem opposet 29372
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 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2283 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2283 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2283 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2283 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2283 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29370 . 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 958 . 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 187 1  |-  ( K  e.  OP  ->  K  e.  Poset )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   Posetcpo 14074   joincjn 14078   meetcmee 14079   0.cp0 14143   1.cp1 14144   OPcops 29362
This theorem is referenced by:  op0le  29376  ople0  29377  ople1  29381  op1le  29382  opltcon3b  29394  olposN  29405  ncvr1  29462  cvrcmp2  29474  leatb  29482  dalemcea  29849
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  ax-nul 4149
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-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-oposet 29366
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