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Theorem tosso 17083
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 17012 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
543expb 1285 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
61, 2, 3pleval2 17012 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑦 = 𝑥)))
7 equcom 1991 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
87orbi2i 540 . . . . . . . . . 10 ((𝑦 < 𝑥𝑦 = 𝑥) ↔ (𝑦 < 𝑥𝑥 = 𝑦))
96, 8syl6bb 276 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
1093com23 1291 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
11103expb 1285 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
125, 11orbi12d 746 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦))))
13 df-3or 1055 . . . . . . 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 264 . . . . . . 7 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1713, 16bitri 264 . . . . . 6 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦)))
1812, 17syl6bbr 278 . . . . 5 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
19182ralbidva 3017 . . . 4 (𝐾 ∈ Poset → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2019pm5.32i 670 . . 3 ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
211, 2, 3pospo 17020 . . . 4 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
2221anbi1d 741 . . 3 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
2320, 22syl5bb 272 . 2 (𝐾𝑉 → ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥))))
241, 2istos 17082 . 2 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
25 df-so 5065 . . . 4 ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2625anbi1i 731 . . 3 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ))
27 an32 856 . . 3 ((( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2826, 27bitri 264 . 2 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2923, 24, 283bitr4g 303 1 (𝐾𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607   class class class wbr 4685   I cid 5052   Po wpo 5062   Or wor 5063  cres 5145  cfv 5926  Basecbs 15904  lecple 15995  Posetcpo 16987  ltcplt 16988  Tosetctos 17080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-preset 16975  df-poset 16993  df-plt 17005  df-toset 17081
This theorem is referenced by:  opsrtoslem2  19533  opsrso  19535  retos  20012  toslub  29796  tosglb  29798  orngsqr  29932
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