Theorem tosso 18340

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 18258	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
5	4	3expb 1120	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ≤ 𝑦 ↔ (𝑥 < 𝑦 ∨ 𝑥 = 𝑦)))
6	1, 2, 3	pleval2 18258	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
7		equcom 2019	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
8	7	orbi2i 912	. . . . . . . . . 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 1126	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
11	10	3expb 1120	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦 ≤ 𝑥 ↔ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
12	5, 11	orbi12d 918	. . . . . 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 925	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15		orordir 929	. . . . . . . 8 ⊢ (((𝑥 < 𝑦 ∨ 𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
16	14, 15	bitri 275	. . . . . . 7 ⊢ (((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦 ∨ 𝑥 = 𝑦) ∨ (𝑦 < 𝑥 ∨ 𝑥 = 𝑦)))
17	13, 16	bitri 275	. . . . . 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 3198	. . . 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 18266	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))
22	21	anbi1d 631	. . 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 283	. 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 18339	. 2 ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
25		df-so 5533	. . . 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 646	. . 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 275	. 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 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901   class class class wbr 5098   I cid 5518   Po wpo 5530   Or wor 5531   ↾ cres 5626  ‘cfv 6492  Basecbs 17136  lecple 17184  Posetcpo 18230  ltcplt 18231  Tosetctos 18337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-iota 6448  df-fun 6494  df-fv 6500  df-proset 18217  df-poset 18236  df-plt 18251  df-toset 18338

This theorem is referenced by:  orngsqr  20799  retos  21573  opsrtoslem2  22011  opsrso  22013  toslub  33055  tosglb  33057