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Theorem pltval3 16291
Description: Alternate expression for less-than relation. (dfpss3 3505 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
pleval2.b  |-  B  =  ( Base `  K
)
pleval2.l  |-  .<_  =  ( le `  K )
pleval2.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltval3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )

Proof of Theorem pltval3
StepHypRef Expression
1 pleval2.l . . 3  |-  .<_  =  ( le `  K )
2 pleval2.s . . 3  |-  .<  =  ( lt `  K )
31, 2pltval 16284 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
4 pleval2.b . . . . . . . . 9  |-  B  =  ( Base `  K
)
54, 1posref 16274 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
653adant3 1050 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
7 breq1 4398 . . . . . . 7  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
86, 7syl5ibcom 228 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
98adantr 472 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  ->  Y  .<_  X ) )
104, 1posasymb 16276 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )
1110biimpd 212 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
1211expdimp 444 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( Y  .<_  X  ->  X  =  Y ) )
139, 12impbid 195 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  <->  Y  .<_  X ) )
1413necon3abid 2679 . . 3  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =/= 
Y  <->  -.  Y  .<_  X ) )
1514pm5.32da 653 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  X  =/=  Y )  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
163, 15bitrd 261 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263   ltcplt 16264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-preset 16251  df-poset 16269  df-plt 16282
This theorem is referenced by:  tltnle  28498  opltcon3b  32841
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