Theorem isprsd 49445

Description: Property of being a preordered set (deduction form). (Contributed by Zhi Wang, 18-Sep-2024.)

Hypotheses

Ref	Expression
isprsd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
isprsd.l	⊢ (𝜑 → ≤ = (le‘𝐾))
isprsd.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
isprsd	⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 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

Step	Hyp	Ref	Expression
1		isprsd.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2	1	elexd 3454	. . 3 ⊢ (𝜑 → 𝐾 ∈ V)
3		eqid 2739	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
4		eqid 2739	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
5	3, 4	isprs 18253	. . . 4 ⊢ (𝐾 ∈ Proset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
6	5	baib 540	. . 3 ⊢ (𝐾 ∈ V → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
7	2, 6	syl 17	. 2 ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
8		isprsd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
9		isprsd.l	. . . . . . 7 ⊢ (𝜑 → ≤ = (le‘𝐾))
10	9	breqd 5083	. . . . . 6 ⊢ (𝜑 → (𝑥 ≤ 𝑥 ↔ 𝑥(le‘𝐾)𝑥))
11	9	breqd 5083	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐾)𝑦))
12	9	breqd 5083	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐾)𝑧))
13	11, 12	anbi12d 638	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧)))
14	9	breqd 5083	. . . . . . 7 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐾)𝑧))
15	13, 14	imbi12d 345	. . . . . 6 ⊢ (𝜑 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
16	10, 15	anbi12d 638	. . . . 5 ⊢ (𝜑 → ((𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
17	8, 16	raleqbidv 3313	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
18	8, 17	raleqbidv 3313	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
19	8, 18	raleqbidv 3313	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
20	7, 19	bitr4d 283	1 ⊢ (𝜑 → (𝐾 ∈ Proset ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431   class class class wbr 5072  ‘cfv 6485  Basecbs 17170  lecple 17218   Proset cproset 18249

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493  df-proset 18251

This theorem is referenced by:  catprs2  49502  prstcprs  50050