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

Theorem isposi 16779
Description: Properties that determine a poset (implicit structure version). (Contributed by NM, 11-Sep-2011.)
Hypotheses
Ref Expression
isposi.k 𝐾 ∈ V
isposi.b 𝐵 = (Base‘𝐾)
isposi.l = (le‘𝐾)
isposi.1 (𝑥𝐵𝑥 𝑥)
isposi.2 ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
isposi.3 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
Assertion
Ref Expression
isposi 𝐾 ∈ Poset
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝐾(𝑥,𝑦,𝑧)

Proof of Theorem isposi
StepHypRef Expression
1 isposi.k . 2 𝐾 ∈ V
2 isposi.1 . . . . 5 (𝑥𝐵𝑥 𝑥)
323ad2ant1 1075 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → 𝑥 𝑥)
4 isposi.2 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
543adant3 1074 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
6 isposi.3 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
73, 5, 63jca 1235 . . 3 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
87rgen3 2959 . 2 𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
9 isposi.b . . 3 𝐵 = (Base‘𝐾)
10 isposi.l . . 3 = (le‘𝐾)
119, 10ispos 16770 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
121, 8, 11mpbir2an 957 1 𝐾 ∈ Poset
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173   class class class wbr 4583  cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-poset 16769
This theorem is referenced by:  isposix  16780  xrstos  29010  xrge0omnd  29042
  Copyright terms: Public domain W3C validator