Theorem pocl

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)

	Ref	Expression
Assertion	pocl	⊢ ( 𝑅 Po 𝐴 → ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		df-po	⊢ ( 𝑅 Po 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
2	1	biimpi	⊢ ( 𝑅 Po 𝐴 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) )
3		id	⊢ ( 𝑥 = 𝐵 → 𝑥 = 𝐵 )
4	3 3	breq12d	⊢ ( 𝑥 = 𝐵 → ( 𝑥 𝑅 𝑥 ↔ 𝐵 𝑅 𝐵 ) )
5	4	notbid	⊢ ( 𝑥 = 𝐵 → ( ¬ 𝑥 𝑅 𝑥 ↔ ¬ 𝐵 𝑅 𝐵 ) )
6		breq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 𝑅 𝑦 ↔ 𝐵 𝑅 𝑦 ) )
7	6	anbi1d	⊢ ( 𝑥 = 𝐵 → ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ↔ ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ) )
8		breq1	⊢ ( 𝑥 = 𝐵 → ( 𝑥 𝑅 𝑧 ↔ 𝐵 𝑅 𝑧 ) )
9	7 8	imbi12d	⊢ ( 𝑥 = 𝐵 → ( ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ↔ ( ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) )
10	5 9	anbi12d	⊢ ( 𝑥 = 𝐵 → ( ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) ↔ ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) ) )
11		breq2	⊢ ( 𝑦 = 𝐶 → ( 𝐵 𝑅 𝑦 ↔ 𝐵 𝑅 𝐶 ) )
12		breq1	⊢ ( 𝑦 = 𝐶 → ( 𝑦 𝑅 𝑧 ↔ 𝐶 𝑅 𝑧 ) )
13	11 12	anbi12d	⊢ ( 𝑦 = 𝐶 → ( ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) ↔ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) ) )
14	13	imbi1d	⊢ ( 𝑦 = 𝐶 → ( ( ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ↔ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) )
15	14	anbi2d	⊢ ( 𝑦 = 𝐶 → ( ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) ↔ ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) ) )
16		breq2	⊢ ( 𝑧 = 𝐷 → ( 𝐶 𝑅 𝑧 ↔ 𝐶 𝑅 𝐷 ) )
17	16	anbi2d	⊢ ( 𝑧 = 𝐷 → ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) ↔ ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) ) )
18		breq2	⊢ ( 𝑧 = 𝐷 → ( 𝐵 𝑅 𝑧 ↔ 𝐵 𝑅 𝐷 ) )
19	17 18	imbi12d	⊢ ( 𝑧 = 𝐷 → ( ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ↔ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) )
20	19	anbi2d	⊢ ( 𝑧 = 𝐷 → ( ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝑧 ) → 𝐵 𝑅 𝑧 ) ) ↔ ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) ) )
21	10 15 20	rspc3v	⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ¬ 𝑥 𝑅 𝑥 ∧ ( ( 𝑥 𝑅 𝑦 ∧ 𝑦 𝑅 𝑧 ) → 𝑥 𝑅 𝑧 ) ) → ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) ) )
22	2 21	syl5com	⊢ ( 𝑅 Po 𝐴 → ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴 ) → ( ¬ 𝐵 𝑅 𝐵 ∧ ( ( 𝐵 𝑅 𝐶 ∧ 𝐶 𝑅 𝐷 ) → 𝐵 𝑅 𝐷 ) ) ) )