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Theorem dfso3 31939
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 4069 . . . . 5 (𝑦𝐴𝐴 ≠ ∅)
2 r19.27zv 4212 . . . . 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 3128 . . 3 (∀𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
54ralbii 3129 . 2 (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
6 df-3an 1073 . . . 4 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
76ralbii 3129 . . 3 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
872ralbii 3130 . 2 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
9 df-po 5170 . . . 4 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109anbi1i 610 . . 3 ((𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
11 df-so 5171 . . 3 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
12 r19.26-2 3213 . . 3 (∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
1310, 11, 123bitr4i 292 . 2 (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴 (∀𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
145, 8, 133bitr4ri 293 1 (𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3o 1070  w3a 1071  wcel 2145  wne 2943  wral 3061  c0 4063   class class class wbr 4786   Po wpo 5168   Or wor 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-v 3353  df-dif 3726  df-nul 4064  df-po 5170  df-so 5171
This theorem is referenced by: (None)
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