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Theorem pltirr 18288
Description: The "less than" relation is not reflexive. (pssirr 4101 analog.) (Contributed by NM, 7-Feb-2012.)
Hypothesis
Ref Expression
pltne.s < = (lt‘𝐾)
Assertion
Ref Expression
pltirr ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)

Proof of Theorem pltirr
StepHypRef Expression
1 eqid 2733 . 2 𝑋 = 𝑋
2 pltne.s . . . . 5 < = (lt‘𝐾)
32pltne 18287 . . . 4 ((𝐾𝐴𝑋𝐵𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
433anidm23 1422 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
54necon2bd 2957 . 2 ((𝐾𝐴𝑋𝐵) → (𝑋 = 𝑋 → ¬ 𝑋 < 𝑋))
61, 5mpi 20 1 ((𝐾𝐴𝑋𝐵) → ¬ 𝑋 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941   class class class wbr 5149  cfv 6544  ltcplt 18261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-plt 18283
This theorem is referenced by:  pospo  18298  atnlt  38183  llnnlt  38394  lplnnlt  38436  lvolnltN  38489
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