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Theorem tosso 16957
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 16886 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
543expb 1263 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 𝑦 ↔ (𝑥 < 𝑦𝑥 = 𝑦)))
61, 2, 3pleval2 16886 . . . . . . . . . 10 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑦 = 𝑥)))
7 equcom 1942 . . . . . . . . . . 11 (𝑦 = 𝑥𝑥 = 𝑦)
87orbi2i 541 . . . . . . . . . 10 ((𝑦 < 𝑥𝑦 = 𝑥) ↔ (𝑦 < 𝑥𝑥 = 𝑦))
96, 8syl6bb 276 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑦𝐵𝑥𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
1093com23 1268 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑦𝐵) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
11103expb 1263 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → (𝑦 𝑥 ↔ (𝑦 < 𝑥𝑥 = 𝑦)))
125, 11orbi12d 745 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 𝑦𝑦 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ (𝑦 < 𝑥𝑥 = 𝑦))))
13 df-3or 1037 . . . . . . 7 ((𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥))
14 or32 549 . . . . . . . 8 (((𝑥 < 𝑦𝑥 = 𝑦) ∨ 𝑦 < 𝑥) ↔ ((𝑥 < 𝑦𝑦 < 𝑥) ∨ 𝑥 = 𝑦))
15 orordir 553 . . . . . . . 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 2982 . . . 4 (𝐾 ∈ Poset → (∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2019pm5.32i 668 . . 3 ((𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)) ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
211, 2, 3pospo 16894 . . . 4 (𝐾𝑉 → (𝐾 ∈ Poset ↔ ( < Po 𝐵 ∧ ( I ↾ 𝐵) ⊆ )))
2221anbi1d 740 . . 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 16956 . 2 (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 𝑥)))
25 df-so 4996 . . . 4 ( < Or 𝐵 ↔ ( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)))
2625anbi1i 730 . . 3 (( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ) ↔ (( < Po 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 < 𝑦𝑥 = 𝑦𝑦 < 𝑥)) ∧ ( I ↾ 𝐵) ⊆ ))
27 an32 838 . . 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 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3555   class class class wbr 4613   I cid 4984   Po wpo 4993   Or wor 4994  cres 5076  cfv 5847  Basecbs 15781  lecple 15869  Posetcpo 16861  ltcplt 16862  Tosetctos 16954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-preset 16849  df-poset 16867  df-plt 16879  df-toset 16955
This theorem is referenced by:  opsrtoslem2  19404  opsrso  19406  retos  19883  toslub  29453  tosglb  29455  orngsqr  29589
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