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Theorem tosso 18465
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 (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))

Proof of Theorem tosso
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tosso.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
2 tosso.l . . . . . . . . 9 = (le‘𝐾)
3 tosso.s . . . . . . . . 9 < = (lt‘𝐾)
41, 2, 3pleval2 18383 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
543expb 1120 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
61, 2, 3pleval2 18383 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑦 = 𝑥)))
7 equcom 2016 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
87orbi2i 912 . . . . . . . . . 10 ((𝑦 < 𝑥𝑦 = 𝑥) ↔ (𝑦 < 𝑥𝑥 = 𝑦))
96, 8bitrdi 287 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
1093com23 1126 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
11103expb 1120 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
125, 11orbi12d 918 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦))))
13 df-3or 1087 . . . . . . 7 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14 or32 925 . . . . . . . 8 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15 orordir 929 . . . . . . . 8 (((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1614, 15bitri 275 . . . . . . 7 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1713, 16bitri 275 . . . . . 6 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1812, 17bitr4di 289 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
19182ralbidva 3218 . . . 4 (𝐾 ∈ Poset → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2019pm5.32i 574 . . 3 ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
211, 2, 3pospo 18391 . . . 4 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
2221anbi1d 631 . . 3 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
2320, 22bitrid 283 . 2 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
241, 2istos 18464 . 2 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
25 df-so 5592 . . . 4 ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2625anbi1i 624 . . 3 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ))
27 an32 646 . . 3 ((( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2826, 27bitri 275 . 2 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2923, 24, 283bitr4g 314 1 (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3o 1085  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wss 3950   class class class wbr 5142   I cid 5576   Po wpo 5589   Or wor 5590  cres 5686  cfv 6560  Basecbs 17248  lecple 17305  Posetcpo 18354  ltcplt 18355  Tosetctos 18462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-po 5591  df-so 5592  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-res 5696  df-iota 6513  df-fun 6562  df-fv 6568  df-proset 18341  df-poset 18360  df-plt 18376  df-toset 18463
This theorem is referenced by:  retos  21637  opsrtoslem2  22081  opsrso  22083  toslub  32964  tosglb  32966  orngsqr  33335
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