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Theorem tosso 17726
 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 17655 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
543expb 1117 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
61, 2, 3pleval2 17655 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑦 = 𝑥)))
7 equcom 2025 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
87orbi2i 910 . . . . . . . . . 10 ((𝑦 < 𝑥𝑦 = 𝑥) ↔ (𝑦 < 𝑥𝑥 = 𝑦))
96, 8bitrdi 290 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
1093com23 1123 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
11103expb 1117 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
125, 11orbi12d 916 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦))))
13 df-3or 1085 . . . . . . 7 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14 or32 923 . . . . . . . 8 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15 orordir 927 . . . . . . . 8 (((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1614, 15bitri 278 . . . . . . 7 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1713, 16bitri 278 . . . . . 6 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1812, 17bitr4di 292 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
19182ralbidva 3127 . . . 4 (𝐾 ∈ Poset → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2019pm5.32i 578 . . 3 ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
211, 2, 3pospo 17663 . . . 4 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
2221anbi1d 632 . . 3 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
2320, 22syl5bb 286 . 2 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
241, 2istos 17725 . 2 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
25 df-so 5448 . . . 4 ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2625anbi1i 626 . . 3 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ))
27 an32 645 . . 3 ((( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2826, 27bitri 278 . 2 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2923, 24, 283bitr4g 317 1 (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860   class class class wbr 5036   I cid 5433   Po wpo 5445   Or wor 5446   ↾ cres 5530  ‘cfv 6340  Basecbs 16555  lecple 16644  Posetcpo 17630  ltcplt 17631  Tosetctos 17723 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-iota 6299  df-fun 6342  df-fv 6348  df-proset 17618  df-poset 17636  df-plt 17648  df-toset 17724 This theorem is referenced by:  retos  20397  opsrtoslem2  20830  opsrso  20832  toslub  30790  tosglb  30792  orngsqr  31042
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