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Theorem pltne 17562
Description: The "less than" relation is not reflexive. (df-pss 3958 analog.) (Contributed by NM, 2-Dec-2011.)
Hypothesis
Ref Expression
pltne.s < = (lt‘𝐾)
Assertion
Ref Expression
pltne ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋𝑌))

Proof of Theorem pltne
StepHypRef Expression
1 eqid 2826 . . . 4 (le‘𝐾) = (le‘𝐾)
2 pltne.s . . . 4 < = (lt‘𝐾)
31, 2pltval 17560 . . 3 ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌 ↔ (𝑋(le‘𝐾)𝑌𝑋𝑌)))
43simplbda 500 . 2 (((𝐾𝐴𝑋𝐵𝑌𝐶) ∧ 𝑋 < 𝑌) → 𝑋𝑌)
54ex 413 1 ((𝐾𝐴𝑋𝐵𝑌𝐶) → (𝑋 < 𝑌𝑋𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  cfv 6352  lecple 16562  ltcplt 17541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-plt 17558
This theorem is referenced by:  pltirr  17563  ogrpaddlt  30635  ornglmullt  30797  orngrmullt  30798  ofldchr  30804  isarchiofld  30807  atlen0  36316  1cvratex  36479  ps-2  36484  lhpn0  37010
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