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Theorem opnoncon 32737
Description: Law of contradiction for orthoposets. (chocin 27140 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 2423 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 opnoncon.o . . . 4  |-  ._|_  =  ( oc `  K )
4 eqid 2423 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 opnoncon.m . . . 4  |-  ./\  =  ( meet `  K )
6 opnoncon.z . . . 4  |-  .0.  =  ( 0. `  K )
7 eqid 2423 . . . 4  |-  ( 1.
`  K )  =  ( 1. `  K
)
81, 2, 3, 4, 5, 6, 7oposlem 32711 . . 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 1324 . 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 1020 1  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  ( X  ./\  (  ._|_  `  X ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   occoc 15191   joincjn 16182   meetcmee 16183   0.cp0 16276   1.cp1 16277   OPcops 32701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-nul 4553
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-dm 4861  df-iota 5563  df-fv 5607  df-ov 6306  df-oposet 32705
This theorem is referenced by:  omlfh1N  32787  omlspjN  32790  atlatmstc  32848  pnonsingN  33461  lhpocnle  33544  dochnoncon  34922
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