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Theorem opposet 29190
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 2316 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2316 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2316 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
4 eqid 2316 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2316 . . 3  |-  ( meet `  K )  =  (
meet `  K )
6 eqid 2316 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
7 eqid 2316 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7isopos 29188 . 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 1633    e. wcel 1701   A.wral 2577   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   occoc 13263   Posetcpo 14123   joincjn 14127   meetcmee 14128   0.cp0 14192   1.cp1 14193   OPcops 29180
This theorem is referenced by:  op0le  29194  ople0  29195  ople1  29199  op1le  29200  opltcon3b  29212  olposN  29223  ncvr1  29280  cvrcmp2  29292  leatb  29300  dalemcea  29667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903  df-oposet 29184
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