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Theorem pltle 14095
Description: Less-than implies less-than-or-equal. (pssss 3271 analog.) (Contributed by NM, 4-Dec-2011.)
Hypotheses
Ref Expression
pltval.l  |-  .<_  =  ( le `  K )
pltval.s  |-  .<  =  ( lt `  K )
Assertion
Ref Expression
pltle  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  ->  X  .<_  Y ) )

Proof of Theorem pltle
StepHypRef Expression
1 pltval.l . . . 4  |-  .<_  =  ( le `  K )
2 pltval.s . . . 4  |-  .<  =  ( lt `  K )
31, 2pltval 14094 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
43simprbda 606 . 2  |-  ( ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C
)  /\  X  .<  Y )  ->  X  .<_  Y )
54ex 423 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 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255   lecple 13215   ltcplt 14075
This theorem is referenced by:  pleval2  14099  pltnlt  14102  pltn2lp  14103  plttr  14104  pospo  14107  atnlt  29503  cvlcvr1  29529  hlrelat  29591  hlrelat3  29601  cvratlem  29610  atltcvr  29624  atlelt  29627  llnnlt  29712  lplnnle2at  29730  lplnnlt  29754  lvolnle3at  29771  lvolnltN  29807  cdlemblem  29982  cdlemb  29983  lhpexle1  30197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-plt 14092
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