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Theorem pltle 14373
Description: Less-than implies less-than-or-equal. (pssss 3402 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 14372 . . 3  |-  ( ( K  e.  A  /\  X  e.  B  /\  Y  e.  C )  ->  ( X  .<  Y  <->  ( X  .<_  Y  /\  X  =/= 
Y ) ) )
43simprbda 607 . 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 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413   lecple 13491   ltcplt 14353
This theorem is referenced by:  pleval2  14377  pltnlt  14380  pltn2lp  14381  plttr  14382  pospo  14385  ofldsqr  24193  ofldaddlt  24194  atnlt  29796  cvlcvr1  29822  hlrelat  29884  hlrelat3  29894  cvratlem  29903  atltcvr  29917  atlelt  29920  llnnlt  30005  lplnnle2at  30023  lplnnlt  30047  lvolnle3at  30064  lvolnltN  30100  cdlemblem  30275  cdlemb  30276  lhpexle1  30490
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-plt 14370
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