Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfso3 Structured version   Visualization version   GIF version

Theorem dfso3 32047
Description: Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
Assertion
Ref Expression
dfso3 (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dfso3
StepHypRef Expression
1 ne0i 4085 . . . . 5 (𝑦𝐴𝐴 ≠ ∅)
2 r19.27zv 4230 . . . . 5 (𝐴 ≠ ∅ → (∀𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
31, 2syl 17 . . . 4 (𝑦𝐴 → (∀𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
43ralbiia 3126 . . 3 (∀𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
54ralbii 3127 . 2 (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
6 df-3an 1109 . . . 4 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
76ralbii 3127 . . 3 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
872ralbii 3128 . 2 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-po 5198 . . . 4 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109anbi1i 617 . . 3 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5199 . . 3 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
12 r19.26-2 3212 . . 3 (∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
1310, 11, 123bitr4i 294 . 2 (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
145, 8, 133bitr4ri 295 1 (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3o 1106  w3a 1107  wcel 2155  wne 2937  wral 3055  c0 4079   class class class wbr 4809   Po wpo 5196   Or wor 5197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-v 3352  df-dif 3735  df-nul 4080  df-po 5198  df-so 5199
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator