Theorem tosso 18052

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 17970	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
5	4	3expb 1118	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
6	1, 2, 3	pleval2 17970	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
7		equcom 2022	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	orbi2i 909	. . . . . . . . . 10 ⊢ ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))
9	6, 8	bitrdi 286	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
10	9	3com23 1124	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
11	10	3expb 1118	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
12	5, 11	orbi12d 915	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦))))
13		df-3or 1086	. . . . . . 7 ⊢ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14		or32 922	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15		orordir 926	. . . . . . . 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 288	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥) ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
19	18	2ralbidva 3121	. . . 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 574	. . 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 17978	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))
22	21	anbi1d 629	. . 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	syl5bb 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 18051	. 2 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
25		df-so 5495	. . . 4 ⊢ ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)))
26	25	anbi1i 623	. . 3 ⊢ (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ ) ↔ (( < Po 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ≤ ))
27		an32 642	. . 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 313	1 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883   class class class wbr 5070   I cid 5479   Po wpo 5492   Or wor 5493   ↾ cres 5582  ‘cfv 6418  Basecbs 16840  lecple 16895  Posetcpo 17940  ltcplt 17941  Tosetctos 18049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-proset 17928  df-poset 17946  df-plt 17963  df-toset 18050

This theorem is referenced by:  retos  20735  opsrtoslem2  21173  opsrso  21175  toslub  31153  tosglb  31155  orngsqr  31405