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Theorem isprsd 45818
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 3420 . . 3 (𝜑𝐾 ∈ V)
3 eqid 2739 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
4 eqid 2739 . . . . 5 (le‘𝐾) = (le‘𝐾)
53, 4isprs 17668 . . . 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 539 . . 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 5051 . . . . . 6 (𝜑 → (𝑥 𝑥𝑥(le‘𝐾)𝑥))
119breqd 5051 . . . . . . . 8 (𝜑 → (𝑥 𝑦𝑥(le‘𝐾)𝑦))
129breqd 5051 . . . . . . . 8 (𝜑 → (𝑦 𝑧𝑦(le‘𝐾)𝑧))
1311, 12anbi12d 634 . . . . . . 7 (𝜑 → ((𝑥 𝑦𝑦 𝑧) ↔ (𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧)))
149breqd 5051 . . . . . . 7 (𝜑 → (𝑥 𝑧𝑥(le‘𝐾)𝑧))
1513, 14imbi12d 348 . . . . . 6 (𝜑 → (((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧) ↔ ((𝑥(le‘𝐾)𝑦𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
1610, 15anbi12d 634 . . . . 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 3305 . . . 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 3305 . . 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 3305 . 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 285 1 (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 (𝑥 𝑥 ∧ ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wral 3054  Vcvv 3400   class class class wbr 5040  cfv 6349  Basecbs 16598  lecple 16687   Proset cproset 17664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5184
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-iota 6307  df-fv 6357  df-proset 17666
This theorem is referenced by:  catprs2  45821  prstcprs  45860
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