This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 GG, 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 ) ) ) )