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Theorem opnoncon 29216
Description: Law of contradiction for orthoposets. (chocin 22129 analog.) (Contributed by NM, 13-Sep-2011.)
Hypotheses
Ref Expression
opnoncon.b  |-  B  =  ( Base `  K
)
opnoncon.o  |-  ._|_  =  ( oc `  K )
opnoncon.m  |-  ./\  =  ( meet `  K )
opnoncon.z  |-  .0.  =  ( 0. `  K )
Assertion
Ref Expression
opnoncon  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X ) )  =  .0.  )

Proof of Theorem opnoncon
StepHypRef Expression
1 opnoncon.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2316 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opnoncon.o . . . 4  |-  ._|_  =  ( oc `  K )
4 eqid 2316 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 opnoncon.m . . . 4  |-  ./\  =  ( meet `  K )
6 opnoncon.z . . . 4  |-  .0.  =  ( 0. `  K )
7 eqid 2316 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 29191 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X  ./\  (  ._|_  `  X )
)  =  .0.  )
)
983anidm23 1241 . 2  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( ( (  ._|_  `  X )  e.  B  /\  (  ._|_  `  (  ._|_  `  X ) )  =  X  /\  ( X ( le `  K ) X  -> 
(  ._|_  `  X )
( le `  K
) (  ._|_  `  X
) ) )  /\  ( X ( join `  K
) (  ._|_  `  X
) )  =  ( 1. `  K )  /\  ( X  ./\  (  ._|_  `  X )
)  =  .0.  )
)
109simp3d 969 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X ) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   occoc 13263   joincjn 14127   meetcmee 14128   0.cp0 14192   1.cp1 14193   OPcops 29180
This theorem is referenced by:  omlfh1N  29266  omlspjN  29269  atlatmstc  29327  pnonsingN  29940  lhpocnle  30023  dochnoncon  31399
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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