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

Theorem ispos 16770
 Description: The predicate "is a poset." (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 4-Nov-2013.)
Hypotheses
Ref Expression
ispos.b 𝐵 = (Base‘𝐾)
ispos.l = (le‘𝐾)
Assertion
Ref Expression
ispos (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥, ,𝑦,𝑧
Allowed substitution hints:   𝐾(𝑥,𝑦,𝑧)

Proof of Theorem ispos
Dummy variables 𝑝 𝑏 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6103 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 ispos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2syl6eqr 2662 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
43eqeq2d 2620 . . . . 5 (𝑝 = 𝐾 → (𝑏 = (Base‘𝑝) ↔ 𝑏 = 𝐵))
5 fveq2 6103 . . . . . . 7 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
6 ispos.l . . . . . . 7 = (le‘𝐾)
75, 6syl6eqr 2662 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = )
87eqeq2d 2620 . . . . 5 (𝑝 = 𝐾 → (𝑟 = (le‘𝑝) ↔ 𝑟 = ))
94, 83anbi12d 1392 . . . 4 (𝑝 = 𝐾 → ((𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ (𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
1092exbidv 1839 . . 3 (𝑝 = 𝐾 → (∃𝑏𝑟(𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ ∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
11 df-poset 16769 . . 3 Poset = {𝑝 ∣ ∃𝑏𝑟(𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
1210, 11elab4g 3324 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
13 fvex 6113 . . . . 5 (Base‘𝐾) ∈ V
142, 13eqeltri 2684 . . . 4 𝐵 ∈ V
15 fvex 6113 . . . . 5 (le‘𝐾) ∈ V
166, 15eqeltri 2684 . . . 4 ∈ V
17 raleq 3115 . . . . . 6 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1817raleqbi1dv 3123 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1918raleqbi1dv 3123 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
20 breq 4585 . . . . . . 7 (𝑟 = → (𝑥𝑟𝑥𝑥 𝑥))
21 breq 4585 . . . . . . . . 9 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
22 breq 4585 . . . . . . . . 9 (𝑟 = → (𝑦𝑟𝑥𝑦 𝑥))
2321, 22anbi12d 743 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥 𝑦𝑦 𝑥)))
2423imbi1d 330 . . . . . . 7 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ↔ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
25 breq 4585 . . . . . . . . 9 (𝑟 = → (𝑦𝑟𝑧𝑦 𝑧))
2621, 25anbi12d 743 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥 𝑦𝑦 𝑧)))
27 breq 4585 . . . . . . . 8 (𝑟 = → (𝑥𝑟𝑧𝑥 𝑧))
2826, 27imbi12d 333 . . . . . . 7 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
2920, 24, 283anbi123d 1391 . . . . . 6 (𝑟 = → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
3029ralbidv 2969 . . . . 5 (𝑟 = → (∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
31302ralbidv 2972 . . . 4 (𝑟 = → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
3214, 16, 19, 31ceqsex2v 3218 . . 3 (∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
3332anbi2i 726 . 2 ((𝐾 ∈ V ∧ ∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))) ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
3412, 33bitri 263 1 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ 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:  ispos2  16771  posi  16773  0pos  16777  isposd  16778  isposi  16779  pospropd  16957  resspos  28990
 Copyright terms: Public domain W3C validator