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Theorem isposi 17164
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 1127 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → 𝑥 𝑥)
4 isposi.2 . . . . 5 ((𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
543adant3 1126 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
6 isposi.3 . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
73, 5, 63jca 1122 . . 3 ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
87rgen3 3125 . 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 17155 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
121, 8, 11mpbir2an 690 1 𝐾 ∈ Poset
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351   class class class wbr 4787  cfv 6030  Basecbs 16064  lecple 16156  Posetcpo 17148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-poset 17154
This theorem is referenced by:  isposix  17165  xrstos  30019  xrge0omnd  30051
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