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Theorem ispos 18254
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 6881 . . . . . . 7 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
2 ispos.b . . . . . . 7 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2791 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
43eqeq2d 2744 . . . . 5 (𝑝 = 𝐾 → (𝑏 = (Base‘𝑝) ↔ 𝑏 = 𝐵))
5 fveq2 6881 . . . . . . 7 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
6 ispos.l . . . . . . 7 = (le‘𝐾)
75, 6eqtr4di 2791 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = )
87eqeq2d 2744 . . . . 5 (𝑝 = 𝐾 → (𝑟 = (le‘𝑝) ↔ 𝑟 = ))
94, 83anbi12d 1438 . . . 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 1928 . . 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 18253 . . 3 Poset = {𝑝 ∣ ∃𝑏𝑟(𝑏 = (Base‘𝑝) ∧ 𝑟 = (le‘𝑝) ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))}
1210, 11elab4g 3671 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)))))
132fvexi 6895 . . . 4 𝐵 ∈ V
146fvexi 6895 . . . 4 ∈ V
15 raleq 3323 . . . . . 6 (𝑏 = 𝐵 → (∀𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1615raleqbi1dv 3334 . . . . 5 (𝑏 = 𝐵 → (∀𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
1716raleqbi1dv 3334 . . . 4 (𝑏 = 𝐵 → (∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))))
18 breq 5146 . . . . . . 7 (𝑟 = → (𝑥𝑟𝑥𝑥 𝑥))
19 breq 5146 . . . . . . . . 9 (𝑟 = → (𝑥𝑟𝑦𝑥 𝑦))
20 breq 5146 . . . . . . . . 9 (𝑟 = → (𝑦𝑟𝑥𝑦 𝑥))
2119, 20anbi12d 632 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥 𝑦𝑦 𝑥)))
2221imbi1d 342 . . . . . . 7 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ↔ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦)))
23 breq 5146 . . . . . . . . 9 (𝑟 = → (𝑦𝑟𝑧𝑦 𝑧))
2419, 23anbi12d 632 . . . . . . . 8 (𝑟 = → ((𝑥𝑟𝑦𝑦𝑟𝑧) ↔ (𝑥 𝑦𝑦 𝑧)))
25 breq 5146 . . . . . . . 8 (𝑟 = → (𝑥𝑟𝑧𝑥 𝑧))
2624, 25imbi12d 345 . . . . . . 7 (𝑟 = → (((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧) ↔ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
2718, 22, 263anbi123d 1437 . . . . . 6 (𝑟 = → ((𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
2827ralbidv 3178 . . . . 5 (𝑟 = → (∀𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
29282ralbidv 3219 . . . 4 (𝑟 = → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
3013, 14, 17, 29ceqsex2v 3529 . . 3 (∃𝑏𝑟(𝑏 = 𝐵𝑟 = ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑥 ∧ ((𝑥𝑟𝑦𝑦𝑟𝑥) → 𝑥 = 𝑦) ∧ ((𝑥𝑟𝑦𝑦𝑟𝑧) → 𝑥𝑟𝑧))) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
3130anbi2i 624 . 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))))
3212, 31bitri 275 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 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wral 3062  Vcvv 3475   class class class wbr 5144  cfv 6535  Basecbs 17131  lecple 17191  Posetcpo 18247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5302
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-iota 6487  df-fv 6543  df-poset 18253
This theorem is referenced by:  ispos2  18255  posi  18257  0pos  18261  0posOLD  18262  isposd  18263  isposi  18264  pospropd  18267  resspos  32107
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