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Theorem pltne 14346
Description: Less-than relation. (df-pss 3279 analog.) (Contributed by NM, 2-Dec-2011.)
Hypothesis
Ref Expression
pltne.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltne  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  ->  X  =/=  Y ) )

Proof of Theorem pltne
StepHypRef Expression
1 eqid 2387 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 pltne.s . . . 4  |-  .<  =  ( lt `  K )
31, 2pltval 14344 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  <->  ( X
( le `  K
) Y  /\  X  =/=  Y ) ) )
43simplbda 608 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C
)  /\  X  .<  Y )  ->  X  =/=  Y )
54ex 424 1  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   class class class wbr 4153   ` cfv 5394   lecple 13463   ltcplt 14325
This theorem is referenced by:  pltirr  14347  ofldaddlt  24067  ofldchr  24070  atlen0  29425  1cvratex  29587  ps-2  29592  lhpn0  30118
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 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-plt 14342
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