Theorem isposd 18041

Description: Properties that determine a poset (implicit structure version). (Contributed by Mario Carneiro, 29-Apr-2014.) (Revised by AV, 26-Apr-2024.)

Hypotheses

Ref	Expression
isposd.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
isposd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
isposd.l	⊢ (𝜑 → ≤ = (le‘𝐾))
isposd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
isposd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
isposd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))

Assertion

Ref	Expression
isposd	⊢ (𝜑 → 𝐾 ∈ Poset)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   ≤ (𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isposd

Step	Hyp	Ref	Expression
1		isposd.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
2	1	elexd 3452	. . 3 ⊢ (𝜑 → 𝐾 ∈ V)
3		isposd.1	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ≤ 𝑥)
4	3	adantrr 714	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ≤ 𝑥)
5	4	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → 𝑥 ≤ 𝑥)
6		isposd.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
7	6	3expb 1119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
8	7	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦))
9		isposd.3	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
10	9	3exp2 1353	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐵 → (𝑧 ∈ 𝐵 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))))
11	10	imp42 427	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))
12	5, 8, 11	3jca 1127	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
13	12	ralrimiva 3103	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
14	13	ralrimivva 3123	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)))
15		isposd.b	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
16		isposd.l	. . . . . . . . 9 ⊢ (𝜑 → ≤ = (le‘𝐾))
17	16	breqd 5085	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ≤ 𝑥 ↔ 𝑥(le‘𝐾)𝑥))
18	16	breqd 5085	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐾)𝑦))
19	16	breqd 5085	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ≤ 𝑥 ↔ 𝑦(le‘𝐾)𝑥))
20	18, 19	anbi12d 631	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) ↔ (𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥)))
21	20	imbi1d 342	. . . . . . . 8 ⊢ (𝜑 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ↔ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦)))
22	16	breqd 5085	. . . . . . . . . 10 ⊢ (𝜑 → (𝑦 ≤ 𝑧 ↔ 𝑦(le‘𝐾)𝑧))
23	18, 22	anbi12d 631	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) ↔ (𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧)))
24	16	breqd 5085	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ≤ 𝑧 ↔ 𝑥(le‘𝐾)𝑧))
25	23, 24	imbi12d 345	. . . . . . . 8 ⊢ (𝜑 → (((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧) ↔ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))
26	17, 21, 25	3anbi123d 1435	. . . . . . 7 ⊢ (𝜑 → ((𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ (𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
27	15, 26	raleqbidv 3336	. . . . . 6 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
28	15, 27	raleqbidv 3336	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧)) ↔ ∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
29	15, 28	raleqbidv 3336	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((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)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
30	29	anbi2d 629	. . 3 ⊢ (𝜑 → ((𝐾 ∈ V ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ≤ 𝑥 ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥) → 𝑥 = 𝑦) ∧ ((𝑥 ≤ 𝑦 ∧ 𝑦 ≤ 𝑧) → 𝑥 ≤ 𝑧))) ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧)))))
31	2, 14, 30	mpbi2and 709	. 2 ⊢ (𝜑 → (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
32		eqid 2738	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
33		eqid 2738	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
34	32, 33	ispos 18032	. 2 ⊢ (𝐾 ∈ Poset ↔ (𝐾 ∈ V ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)∀𝑧 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑥 ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑥) → 𝑥 = 𝑦) ∧ ((𝑥(le‘𝐾)𝑦 ∧ 𝑦(le‘𝐾)𝑧) → 𝑥(le‘𝐾)𝑧))))
35	31, 34	sylibr 233	1 ⊢ (𝜑 → 𝐾 ∈ Poset)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   class class class wbr 5074  ‘cfv 6433  Basecbs 16912  lecple 16969  Posetcpo 18025

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-poset 18031

This theorem is referenced by:  odupos  18046  pospo  18063  ipopos  18254  zntoslem  20764