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Theorem isposd 17221
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 708 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝑥 𝑥)
43adantr 472 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → 𝑥 𝑥)
5 isposd.2 . . . . . . . 8 ((𝜑𝑥𝐵𝑦𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
653expb 1149 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
76adantr 472 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦))
8 isposd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
983exp2 1463 . . . . . . 7 (𝜑 → (𝑥𝐵 → (𝑦𝐵 → (𝑧𝐵 → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))))
109imp42 417 . . . . . 6 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))
114, 7, 103jca 1158 . . . . 5 (((𝜑 ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑧𝐵) → (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1211ralrimiva 3113 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → ∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)))
1312ralrimivva 3118 . . 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 4820 . . . . . . . 8 (𝜑 → (𝑥 𝑥𝑥(le‘𝐾)𝑥))
1715breqd 4820 . . . . . . . . . 10 (𝜑 → (𝑥 𝑦𝑥(le‘𝐾)𝑦))
1815breqd 4820 . . . . . . . . . 10 (𝜑 → (𝑦 𝑥𝑦(le‘𝐾)𝑥))
1917, 18anbi12d 624 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥)))
2019imbi1d 332 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
2115breqd 4820 . . . . . . . . . 10 (𝜑 → (𝑦 𝑧𝑦(le‘𝐾)𝑧))
2217, 21anbi12d 624 . . . . . . . . 9 (𝜑 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧)))
2315breqd 4820 . . . . . . . . 9 (𝜑 → (𝑥 𝑧𝑥(le‘𝐾)𝑧))
2422, 23imbi12d 335 . . . . . . . 8 (𝜑 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
2516, 20, 243anbi123d 1560 . . . . . . 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 3300 . . . . . 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 3300 . . . . 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 3300 . . . 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 622 . . 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 703 . 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 2765 . . 3 (Base‘𝐾) = (Base‘𝐾)
32 eqid 2765 . . 3 (le‘𝐾) = (le‘𝐾)
3331, 32ispos 17213 . 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 225 1 (𝜑𝐾 ∈ Poset)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4809  cfv 6068  Basecbs 16130  lecple 16221  Posetcpo 17206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4949
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-iota 6031  df-fv 6076  df-poset 17212
This theorem is referenced by:  pospo  17239  odupos  17401  ipopos  17426  zntoslem  20177
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