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Theorem pltval3 14424
Description: Alternate expression for less-than relation. (dfpss3 3433 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 14417 . 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 14408 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B )  ->  X  .<_  X )
653adant3 977 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
7 breq1 4215 . . . . . . 7  |-  ( X  =  Y  ->  ( X  .<_  X  <->  Y  .<_  X ) )
86, 7syl5ibcom 212 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  ->  Y 
.<_  X ) )
98adantr 452 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  ->  Y  .<_  X ) )
104, 1posasymb 14409 . . . . . . 7  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  <->  X  =  Y ) )
1110biimpd 199 . . . . . 6  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  Y  .<_  X )  ->  X  =  Y )
)
1211expdimp 427 . . . . 5  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( Y  .<_  X  ->  X  =  Y ) )
139, 12impbid 184 . . . 4  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =  Y  <->  Y  .<_  X ) )
1413necon3abid 2634 . . 3  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<_  Y )  ->  ( X  =/= 
Y  <->  -.  Y  .<_  X ) )
1514pm5.32da 623 . 2  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  (
( X  .<_  Y  /\  X  =/=  Y )  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
163, 15bitrd 245 1  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  -.  Y  .<_  X ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   Posetcpo 14397   ltcplt 14398
This theorem is referenced by:  tltnle  24190  opltcon3b  30002
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-poset 14403  df-plt 14415
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