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Theorem isprsd 48673
Description: Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
isprsd.b (𝜑𝐵 = (Base‘𝐾))
isprsd.l (𝜑 = (le‘𝐾))
isprsd.k (𝜑𝐾𝑉)
Assertion
Ref Expression
isprsd (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧)   (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isprsd
StepHypRef Expression
1 isprsd.k . . . 4 (𝜑𝐾𝑉)
21elexd 3501 . . 3 (𝜑𝐾 ∈ V)
3 eqid 2733 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
4 eqid 2733 . . . . 5 (le‘𝐾) = (le‘𝐾)
53, 4isprs 18343 . . . 4 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
65baib 535 . . 3 (𝐾 ∈ V → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
72, 6syl 17 . 2 (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
8 isprsd.b . . 3 (𝜑𝐵 = (Base‘𝐾))
9 isprsd.l . . . . . . 7 (𝜑 = (le‘𝐾))
109breqd 5160 . . . . . 6 (𝜑 → (𝑥 𝑥𝑥(le‘𝐾)𝑥))
119breqd 5160 . . . . . . . 8 (𝜑 → (𝑥 𝑦𝑥(le‘𝐾)𝑦))
129breqd 5160 . . . . . . . 8 (𝜑 → (𝑦 𝑧𝑦(le‘𝐾)𝑧))
1311, 12anbi12d 631 . . . . . . 7 (𝜑 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧)))
149breqd 5160 . . . . . . 7 (𝜑 → (𝑥 𝑧𝑥(le‘𝐾)𝑧))
1513, 14imbi12d 344 . . . . . 6 (𝜑 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
1610, 15anbi12d 631 . . . . 5 (𝜑 → ((𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
178, 16raleqbidv 3342 . . . 4 (𝜑 → (∀𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
188, 17raleqbidv 3342 . . 3 (𝜑 → (∀𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
198, 18raleqbidv 3342 . 2 (𝜑 → (∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
207, 19bitr4d 282 1 (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  wral 3057  Vcvv 3477   class class class wbr 5149  cfv 6558  Basecbs 17234  lecple 17294   Proset cproset 18339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-iota 6510  df-fv 6566  df-proset 18341
This theorem is referenced by:  catprs2  48722  prstcprs  48796
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