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Theorem tosso 16859
 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 16788 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
543expb 1258 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
61, 2, 3pleval2 16788 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑦 = 𝑥)))
7 equcom 1932 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
87orbi2i 540 . . . . . . . . . 10 ((𝑦 < 𝑥𝑦 = 𝑥) ↔ (𝑦 < 𝑥𝑥 = 𝑦))
96, 8syl6bb 275 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
1093com23 1263 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
11103expb 1258 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
125, 11orbi12d 742 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦))))
13 df-3or 1032 . . . . . . 7 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14 or32 548 . . . . . . . 8 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15 orordir 552 . . . . . . . 8 (((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1614, 15bitri 263 . . . . . . 7 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1713, 16bitri 263 . . . . . 6 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1812, 17syl6bbr 277 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
19182ralbidva 2971 . . . 4 (𝐾 ∈ Poset → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2019pm5.32i 667 . . 3 ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
211, 2, 3pospo 16796 . . . 4 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
2221anbi1d 737 . . 3 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
2320, 22syl5bb 271 . 2 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
241, 2istos 16858 . 2 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
25 df-so 4960 . . . 4 ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2625anbi1i 727 . . 3 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ))
27 an32 835 . . 3 ((( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2826, 27bitri 263 . 2 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2923, 24, 283bitr4g 302 1 (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583   I cid 4948   Po wpo 4957   Or wor 4958   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764  Tosetctos 16856 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-toset 16857 This theorem is referenced by:  opsrtoslem2  19306  opsrso  19308  retos  19783  toslub  28999  tosglb  29001  orngsqr  29135
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