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Theorem oplecon1b 29460
Description: Contraposition law for strict ordering in orthoposets. (chsscon1 22194 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 29453 . . . 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 29459 . . 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 29454 . . . 4  |-  ( ( K  e.  OP  /\  X  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
983adant3 975 . . 3  |-  ( ( K  e.  OP  /\  X  e.  B  /\  Y  e.  B )  ->  (  ._|_  `  (  ._|_  `  X ) )  =  X )
109breq2d 4116 . 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 1642    e. wcel 1710   class class class wbr 4104   ` cfv 5337   Basecbs 13245   lecple 13312   occoc 13313   OPcops 29431
This theorem is referenced by:  opoc1  29461  oldmm1  29476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-ov 5948  df-oposet 29435
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