Theorem pocl 5475

Description: Characteristic properties of a partial order in class notation. (Contributed by NM, 27-Mar-1997.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.)

Assertion

Ref	Expression
pocl	⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Proof of Theorem pocl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-po 5468	. . 3 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
2	1	biimpi 219	. 2 ⊢ (𝑅 Po 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
3		id 22	. . . . . 6 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
4	3, 3	breq12d 5066	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑥 ↔ 𝐵𝑅𝐵))
5	4	notbid 321	. . . 4 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
6		breq1 5056	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
7	6	anbi1d 633	. . . . 5 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧)))
8		breq1 5056	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
9	7, 8	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝐵 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))
10	5, 9	anbi12d 634	. . 3 ⊢ (𝑥 = 𝐵 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))))
11		breq2 5057	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
12		breq1 5056	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝐶𝑅𝑧))
13	11, 12	anbi12d 634	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧)))
14	13	imbi1d 345	. . . 4 ⊢ (𝑦 = 𝐶 → (((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))
15	14	anbi2d 632	. . 3 ⊢ (𝑦 = 𝐶 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))))
16		breq2 5057	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝐷))
17	16	anbi2d 632	. . . . 5 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷)))
18		breq2 5057	. . . . 5 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
19	17, 18	imbi12d 348	. . . 4 ⊢ (𝑧 = 𝐷 → (((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
20	19	anbi2d 632	. . 3 ⊢ (𝑧 = 𝐷 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
21	10, 15, 20	rspc3v 3550	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
22	2, 21	syl5com 31	1 ⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061   class class class wbr 5053   Po wpo 5466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-po 5468

This theorem is referenced by:  poirr  5480  potr  5481