Theorem tosso 18146

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 18064	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
5	4	3expb 1119	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
6	1, 2, 3	pleval2 18064	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
7		equcom 2022	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	orbi2i 910	. . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
9	6, 8	bitrdi 287	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
10	9	3com23 1125	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
11	10	3expb 1119	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
12	5, 11	orbi12d 916	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))))
13		df-3or 1087	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14		or32 923	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15		orordir 927	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
16	14, 15	bitri 274	. . . . . . 7 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
17	13, 16	bitri 274	. . . . . 6 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
18	12, 17	bitr4di 289	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
19	18	2ralbidva 3129	. . . 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 575	. . 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 18072	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))
22	21	anbi1d 630	. . 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 282	. 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 18145	. 2 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
25		df-so 5505	. . . 4 ⊢ ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
26	25	anbi1i 624	. . 3 ⊢ (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ↔ (( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ≤ ))
27		an32 643	. . 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 274	. 2 ⊢ (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
29	23, 24, 28	3bitr4g 314	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3065   ⊆ wss 3888   class class class wbr 5075   I cid 5489   Po wpo 5502   Or wor 5503   ↾ cres 5592  ‘cfv 6437  Basecbs 16921  lecple 16978  Posetcpo 18034  ltcplt 18035  Tosetctos 18143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-sbc 3718  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-id 5490  df-po 5504  df-so 5505  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-res 5602  df-iota 6395  df-fun 6439  df-fv 6445  df-proset 18022  df-poset 18040  df-plt 18057  df-toset 18144

This theorem is referenced by:  retos  20832  opsrtoslem2  21272  opsrso  21274  toslub  31260  tosglb  31262  orngsqr  31512