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Theorem pltnlt 14345
Description: The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
pltnlt.b  |-  B  =  ( Base `  K
)
pltnlt.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltnlt  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<  X )

Proof of Theorem pltnlt
StepHypRef Expression
1 pltnlt.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2380 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 pltnlt.s . . 3  |-  .<  =  ( lt `  K )
41, 2, 3pltnle 14343 . 2  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y ( le `  K ) X )
52, 3pltle 14338 . . . 4  |-  ( ( K  e.  Poset  /\  Y  e.  B  /\  X  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
653com23 1159 . . 3  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  ->  ( Y  .<  X  ->  Y
( le `  K
) X ) )
76adantr 452 . 2  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  ( Y  .<  X  ->  Y ( le
`  K ) X ) )
84, 7mtod 170 1  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  -.  Y  .<  X )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4146   ` cfv 5387   Basecbs 13389   lecple 13456   Posetcpo 14317   ltcplt 14318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-poset 14323  df-plt 14335
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