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Theorem pltne 14411
Description: Less-than relation. (df-pss 3328 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 2435 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 pltne.s . . . 4  |-  .<  =  ( lt `  K )
31, 2pltval 14409 . . 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 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446   lecple 13528   ltcplt 14390
This theorem is referenced by:  pltirr  14412  ofldaddlt  24233  ofldchr  24236  atlen0  30045  1cvratex  30207  ps-2  30212  lhpn0  30738
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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-plt 14407
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