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Theorem isprsd 47588
Description: Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)
Hypotheses
Ref Expression
isprsd.b (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
isprsd.l (πœ‘ β†’ ≀ = (leβ€˜πΎ))
isprsd.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
Assertion
Ref Expression
isprsd (πœ‘ β†’ (𝐾 ∈ Proset ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Distinct variable groups:   π‘₯,𝐾,𝑦,𝑧   πœ‘,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐡(π‘₯,𝑦,𝑧)   ≀ (π‘₯,𝑦,𝑧)   𝑉(π‘₯,𝑦,𝑧)

Proof of Theorem isprsd
StepHypRef Expression
1 isprsd.k . . . 4 (πœ‘ β†’ 𝐾 ∈ 𝑉)
21elexd 3495 . . 3 (πœ‘ β†’ 𝐾 ∈ V)
3 eqid 2733 . . . . 5 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
4 eqid 2733 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
53, 4isprs 18250 . . . 4 (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
65baib 537 . . 3 (𝐾 ∈ V β†’ (𝐾 ∈ Proset ↔ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
72, 6syl 17 . 2 (πœ‘ β†’ (𝐾 ∈ Proset ↔ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
8 isprsd.b . . 3 (πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))
9 isprsd.l . . . . . . 7 (πœ‘ β†’ ≀ = (leβ€˜πΎ))
109breqd 5160 . . . . . 6 (πœ‘ β†’ (π‘₯ ≀ π‘₯ ↔ π‘₯(leβ€˜πΎ)π‘₯))
119breqd 5160 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ≀ 𝑦 ↔ π‘₯(leβ€˜πΎ)𝑦))
129breqd 5160 . . . . . . . 8 (πœ‘ β†’ (𝑦 ≀ 𝑧 ↔ 𝑦(leβ€˜πΎ)𝑧))
1311, 12anbi12d 632 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) ↔ (π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧)))
149breqd 5160 . . . . . . 7 (πœ‘ β†’ (π‘₯ ≀ 𝑧 ↔ π‘₯(leβ€˜πΎ)𝑧))
1513, 14imbi12d 345 . . . . . 6 (πœ‘ β†’ (((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧) ↔ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧)))
1610, 15anbi12d 632 . . . . 5 (πœ‘ β†’ ((π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)) ↔ (π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
178, 16raleqbidv 3343 . . . 4 (πœ‘ β†’ (βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)) ↔ βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
188, 17raleqbidv 3343 . . 3 (πœ‘ β†’ (βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)) ↔ βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
198, 18raleqbidv 3343 . 2 (πœ‘ β†’ (βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧)) ↔ βˆ€π‘₯ ∈ (Baseβ€˜πΎ)βˆ€π‘¦ ∈ (Baseβ€˜πΎ)βˆ€π‘§ ∈ (Baseβ€˜πΎ)(π‘₯(leβ€˜πΎ)π‘₯ ∧ ((π‘₯(leβ€˜πΎ)𝑦 ∧ 𝑦(leβ€˜πΎ)𝑧) β†’ π‘₯(leβ€˜πΎ)𝑧))))
207, 19bitr4d 282 1 (πœ‘ β†’ (𝐾 ∈ Proset ↔ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (π‘₯ ≀ π‘₯ ∧ ((π‘₯ ≀ 𝑦 ∧ 𝑦 ≀ 𝑧) β†’ π‘₯ ≀ 𝑧))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  Vcvv 3475   class class class wbr 5149  β€˜cfv 6544  Basecbs 17144  lecple 17204   Proset cproset 18246
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-proset 18248
This theorem is referenced by:  catprs2  47632  prstcprs  47695
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