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Theorem isposd 18041
Description: Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by AV, 26-Apr-2024.)
Hypotheses
Ref Expression
isposd.k (𝜑𝐾𝑉)
isposd.b (𝜑𝐵 = (Base‘𝐾))
isposd.l (𝜑 = (le‘𝐾))
isposd.1 ((𝜑𝑥𝐵) → 𝑥 𝑥)
isposd.2 ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
isposd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
Assertion
Ref Expression
isposd (𝜑𝐾 ∈ Poset)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isposd
StepHypRef Expression
1 isposd.k . . . 4 (𝜑𝐾𝑉)
21elexd 3452 . . 3 (𝜑𝐾 ∈ V)
3 isposd.1 . . . . . . . 8 ((𝜑𝑥𝐵) → 𝑥 𝑥)
43adantrr 714 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 𝑥)
54adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥 𝑥)
6 isposd.2 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
763expb 1119 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
87adantr 481 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
9 isposd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
1093exp2 1353 . . . . . . 7 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))))
1110imp42 427 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
125, 8, 113jca 1127 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1312ralrimiva 3103 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1413ralrimivva 3123 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
15 isposd.b . . . . 5 (𝜑𝐵 = (Base‘𝐾))
16 isposd.l . . . . . . . . 9 (𝜑 = (le‘𝐾))
1716breqd 5085 . . . . . . . 8 (𝜑 → (𝑥 𝑥𝑥(le‘𝐾)𝑥))
1816breqd 5085 . . . . . . . . . 10 (𝜑 → (𝑥 𝑦𝑥(le‘𝐾)𝑦))
1916breqd 5085 . . . . . . . . . 10 (𝜑 → (𝑦 𝑥𝑦(le‘𝐾)𝑥))
2018, 19anbi12d 631 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥)))
2120imbi1d 342 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
2216breqd 5085 . . . . . . . . . 10 (𝜑 → (𝑦 𝑧𝑦(le‘𝐾)𝑧))
2318, 22anbi12d 631 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧)))
2416breqd 5085 . . . . . . . . 9 (𝜑 → (𝑥 𝑧𝑥(le‘𝐾)𝑧))
2523, 24imbi12d 345 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
2617, 21, 253anbi123d 1435 . . . . . . 7 (𝜑 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
2715, 26raleqbidv 3336 . . . . . 6 (𝜑 → (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
2815, 27raleqbidv 3336 . . . . 5 (𝜑 → (∀𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
2915, 28raleqbidv 3336 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
3029anbi2d 629 . . 3 (𝜑 → ((𝐾 ∈ V ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))) ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))))
312, 14, 30mpbi2and 709 . 2 (𝜑 → (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
32 eqid 2738 . . 3 (Base‘𝐾) = (Base‘𝐾)
33 eqid 2738 . . 3 (le‘𝐾) = (le‘𝐾)
3432, 33ispos 18032 . 2 (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
3531, 34sylibr 233 1 (𝜑𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432   class class class wbr 5074  cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-poset 18031
This theorem is referenced by:  odupos  18046  pospo  18063  ipopos  18254  zntoslem  20764
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