Theorem tosso 18596

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

Hypotheses

Ref	Expression
tosso.b	⊢ 𝐵 = (Base‘𝐾)
tosso.l	⊢ ≤ = (le‘𝐾)
tosso.s	⊢ < = (lt‘𝐾)

Assertion

Ref	Expression
tosso	⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

Proof of Theorem tosso

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

Step	Hyp	Ref	Expression
1		tosso.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
2		tosso.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
3		tosso.s	. . . . . . . . 9 ⊢ < = (lt‘𝐾)
4	1, 2, 3	pleval2 18514	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
5	4	3expb 1138	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
6	1, 2, 3	pleval2 18514	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
7		equcom 2050	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	orbi2i 926	. . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
9	6, 8	bitrdi 290	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
10	9	3com23 1144	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
11	10	3expb 1138	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
12	5, 11	orbi12d 932	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))))
13		df-3or 1104	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14		or32 939	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15		orordir 943	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
16	14, 15	bitri 278	. . . . . . 7 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
17	13, 16	bitri 278	. . . . . 6 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
18	12, 17	bitr4di 292	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
19	18	2ralbidva 3224	. . . 4 ⊢ (𝐾 ∈ Poset → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
20	19	pm5.32i 585	. . 3 ⊢ ((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 18522	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))
22	21	anbi1d 643	. . 3 ⊢ (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	bitrid 286	. 2 ⊢ (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 18595	. 2 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
25		df-so 5559	. . . 4 ⊢ ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
26	25	anbi1i 636	. . 3 ⊢ (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ↔ (( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ≤ ))
27		an32 659	. . 3 ⊢ ((( < 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 278	. 2 ⊢ (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
29	23, 24, 28	3bitr4g 317	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   class class class wbr 5102   I cid 5544   Po wpo 5556   Or wor 5557   ↾ cres 5652  ‘cfv 6538  Basecbs 17392  lecple 17440  Posetcpo 18486  ltcplt 18487  Tosetctos 18593

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-po 5558  df-so 5559  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fv 6546  df-proset 18473  df-poset 18492  df-plt 18507  df-toset 18594

This theorem is used by:  orngsqr  21128  retos  21929  opsrtoslem2  22370  opsrso  22372  toslub  33580  tosglb  33582