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Theorem pltirr 18270
Description: The "less than" relation is not reflexive. (pssirr 4096 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 2731 . 2 𝑋 = 𝑋
2 pltne.s . . . . 5 < = (lt‘𝐾)
32pltne 18269 . . . 4 ((𝐾𝐴𝑋𝐵𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
433anidm23 1421 . . 3 ((𝐾𝐴𝑋𝐵) → (𝑋 < 𝑋𝑋𝑋))
54necon2bd 2955 . 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 396   = wceq 1541  wcel 2106  wne 2939   class class class wbr 5141  cfv 6532  ltcplt 18243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-plt 18265
This theorem is referenced by:  pospo  18280  atnlt  37986  llnnlt  38197  lplnnlt  38239  lvolnltN  38292
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