Metamath Proof Explorer

Theorem tosso

Description: Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015)

Ref Expression
Hypotheses tosso.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
tosso.l ⊢ ≤ = ( le ‘ 𝐾 )
tosso.s ⊢ < = ( lt ‘ 𝐾 )
Assertion tosso ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) )

Proof

Step Hyp Ref Expression
1 tosso.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 tosso.l ⊢ ≤ = ( le ‘ 𝐾 )
3 tosso.s ⊢ < = ( lt ‘ 𝐾 )
4 1 2 3 pleval2 ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) )
5 4 3expb ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) )
6 1 2 3 pleval2 ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
7 equcom ⊢ ( 𝑦 = 𝑥 ↔ 𝑥 = 𝑦 )
8 7 orbi2i ⊢ ( ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) )
9 6 8 bitrdi ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
10 9 3com23 ⊢ ( ( 𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
11 10 3expb ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
12 5 11 orbi12d ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) ) )
13 df-3or ⊢ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) )
14 or32 ⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ) ∨ 𝑥 = 𝑦 ) )
15 orordir ⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑦 < 𝑥 ) ∨ 𝑥 = 𝑦 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
16 14 15 bitri ⊢ ( ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
17 13 16 bitri ⊢ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ↔ ( ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ∨ ( 𝑦 < 𝑥 ∨ 𝑥 = 𝑦 ) ) )
18 12 17 bitr4di ⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
19 18 2ralbidva ⊢ ( 𝐾 ∈ Poset → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
20 19 pm5.32i ⊢ ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ↔ ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
21 1 2 3 pospo ⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) )
22 21 anbi1d ⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) )
23 20 22 syl5bb ⊢ ( 𝐾 ∈ 𝑉 → ( ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ) )
24 1 2 istos ⊢ ( 𝐾 ∈ Toset ↔ ( 𝐾 ∈ Poset ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥 ) ) )
25 df-so ⊢ ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
26 25 anbi1i ⊢ ( ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) )
27 an32 ⊢ ( ( ( < Po 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
28 26 27 bitri ⊢ ( ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ↔ ( ( < Po 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥 ) ) )
29 23 24 28 3bitr4g ⊢ ( 𝐾 ∈ 𝑉 → ( 𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵 ) ⊆ ≤ ) ) )