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Theorem oplecon1b 29391
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 22080 analog.) (Contributed by NM, 6-Nov-2011.)
Hypotheses
Ref Expression
opcon3.b  |-  B  =  ( Base `  K
)
opcon3.l  |-  .<_  =  ( le `  K )
opcon3.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
oplecon1b  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  .<_  Y  <->  (  ._|_  `  Y )  .<_  X ) )

Proof of Theorem oplecon1b
StepHypRef Expression
1 opcon3.b . . . . 5  |-  B  =  ( Base `  K
)
2 opcon3.o . . . . 5  |-  ._|_  =  ( oc `  K )
31, 2opoccl 29384 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
433adant3 975 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
5 opcon3.l . . . 4  |-  .<_  =  ( le `  K )
61, 5, 2oplecon3b 29390 . . 3  |-  ( ( K  e.  OP  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  .<_  Y  <->  (  ._|_  `  Y
)  .<_  (  ._|_  `  (  ._|_  `  X ) ) ) )
74, 6syld3an2 1229 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  .<_  Y  <->  (  ._|_  `  Y )  .<_  (  ._|_  `  (  ._|_  `  X ) ) ) )
81, 2opococ 29385 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
983adant3 975 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
109breq2d 4035 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  Y
)  .<_  (  ._|_  `  (  ._|_  `  X ) )  <-> 
(  ._|_  `  Y )  .<_  X ) )
117, 10bitrd 244 1  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  .<_  Y  <->  (  ._|_  `  Y )  .<_  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255   Basecbs 13148   lecple 13215   occoc 13216   OPcops 29362
This theorem is referenced by:  opoc1  29392  oldmm1  29407
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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