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Theorem oplecon1b 29900
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 22993 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 29893 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  X )  e.  B )
433adant3 977 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  X )  e.  B )
5 opcon3.l . . . 4  |-  .<_  =  ( le `  K )
61, 5, 2oplecon3b 29899 . . 3  |-  ( ( K  e.  OP  /\  (  ._|_  `  X )  e.  B  /\  Y  e.  B )  ->  (
(  ._|_  `  X )  .<_  Y  <->  (  ._|_  `  Y
)  .<_  (  ._|_  `  (  ._|_  `  X ) ) ) )
74, 6syld3an2 1231 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  X
)  .<_  Y  <->  (  ._|_  `  Y )  .<_  (  ._|_  `  (  ._|_  `  X ) ) ) )
81, 2opococ 29894 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
983adant3 977 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
109breq2d 4216 . 2  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  ( (  ._|_  `  Y
)  .<_  (  ._|_  `  (  ._|_  `  X ) )  <-> 
(  ._|_  `  Y )  .<_  X ) )
117, 10bitrd 245 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 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446   Basecbs 13459   lecple 13526   occoc 13527   OPcops 29871
This theorem is referenced by:  opoc1  29901  oldmm1  29916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-oposet 29875
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