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Theorem isposd 17002
Description: Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.)
Hypotheses
Ref Expression
isposd.k (𝜑𝐾 ∈ V)
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)

Proof of Theorem isposd
StepHypRef Expression
1 isposd.k . . 3 (𝜑𝐾 ∈ V)
2 isposd.1 . . . . . . . 8 ((𝜑𝑥𝐵) → 𝑥 𝑥)
32adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 𝑥)
43adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥 𝑥)
5 isposd.2 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
653expb 1285 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
76adantr 480 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
8 isposd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
983exp2 1307 . . . . . . 7 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))))
109imp42 619 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
114, 7, 103jca 1261 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1211ralrimiva 2995 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1312ralrimivva 3000 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
14 isposd.b . . . . 5 (𝜑𝐵 = (Base‘𝐾))
15 isposd.l . . . . . . . . 9 (𝜑 = (le‘𝐾))
1615breqd 4696 . . . . . . . 8 (𝜑 → (𝑥 𝑥𝑥(le‘𝐾)𝑥))
1715breqd 4696 . . . . . . . . . 10 (𝜑 → (𝑥 𝑦𝑥(le‘𝐾)𝑦))
1815breqd 4696 . . . . . . . . . 10 (𝜑 → (𝑦 𝑥𝑦(le‘𝐾)𝑥))
1917, 18anbi12d 747 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥)))
2019imbi1d 330 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
2115breqd 4696 . . . . . . . . . 10 (𝜑 → (𝑦 𝑧𝑦(le‘𝐾)𝑧))
2217, 21anbi12d 747 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧)))
2315breqd 4696 . . . . . . . . 9 (𝜑 → (𝑥 𝑧𝑥(le‘𝐾)𝑧))
2422, 23imbi12d 333 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
2516, 20, 243anbi123d 1439 . . . . . . 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))))
2614, 25raleqbidv 3182 . . . . . 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))))
2714, 26raleqbidv 3182 . . . . 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))))
2814, 27raleqbidv 3182 . . . 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))))
2928anbi2d 740 . . 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)))))
301, 13, 29mpbi2and 976 . 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))))
31 eqid 2651 . . 3 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2651 . . 3 (le‘𝐾) = (le‘𝐾)
3331, 32ispos 16994 . 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))))
3430, 33sylibr 224 1 (𝜑𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231   class class class wbr 4685  cfv 5926  Basecbs 15904  lecple 15995  Posetcpo 16987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-poset 16993
This theorem is referenced by:  pospo  17020  odupos  17182  ipopos  17207  zntoslem  19953
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