MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pltnlt Structured version   Visualization version   GIF version

Theorem pltnlt 16884
Description: The less-than relation implies the negation of its inverse. (Contributed by NM, 18-Oct-2011.)
Hypotheses
Ref Expression
pltnlt.b 𝐵 = (Base‘𝐾)
pltnlt.s < = (lt‘𝐾)
Assertion
Ref Expression
pltnlt (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 < 𝑋)

Proof of Theorem pltnlt
StepHypRef Expression
1 pltnlt.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2626 . . 3 (le‘𝐾) = (le‘𝐾)
3 pltnlt.s . . 3 < = (lt‘𝐾)
41, 2, 3pltnle 16882 . 2 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋)
52, 3pltle 16877 . . . 4 ((𝐾 ∈ Poset ∧ 𝑌𝐵𝑋𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
653com23 1268 . . 3 ((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
76adantr 481 . 2 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑋𝑌(le‘𝐾)𝑋))
84, 7mtod 189 1 (((𝐾 ∈ Poset ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 < 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992   class class class wbr 4618  cfv 5850  Basecbs 15776  lecple 15864  Posetcpo 16856  ltcplt 16857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5813  df-fun 5852  df-fv 5858  df-preset 16844  df-poset 16862  df-plt 16874
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator