Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfso2 Structured version   Visualization version   GIF version

Theorem dfso2 30897
 Description: Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
Assertion
Ref Expression
dfso2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))

Proof of Theorem dfso2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-so 4960 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2 opelxp 5070 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
3 brun 4633 . . . . . . . . . 10 (𝑥( I ∪ 𝑅)𝑦 ↔ (𝑥 I 𝑦𝑥𝑅𝑦))
4 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
54ideq 5196 . . . . . . . . . . 11 (𝑥 I 𝑦𝑥 = 𝑦)
6 vex 3176 . . . . . . . . . . . 12 𝑥 ∈ V
76, 4brcnv 5227 . . . . . . . . . . 11 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 7orbi12i 542 . . . . . . . . . 10 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑦𝑅𝑥))
93, 8bitr2i 264 . . . . . . . . 9 ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥( I ∪ 𝑅)𝑦)
109orbi2i 540 . . . . . . . 8 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
11 3orass 1034 . . . . . . . 8 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
12 brun 4633 . . . . . . . 8 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
1310, 11, 123bitr4i 291 . . . . . . 7 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦)
14 df-br 4584 . . . . . . 7 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))
1513, 14bitr2i 264 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
162, 15imbi12i 339 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
17162albii 1738 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 relxp 5150 . . . . 5 Rel (𝐴 × 𝐴)
19 ssrel 5130 . . . . 5 (Rel (𝐴 × 𝐴) → ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))))
2018, 19ax-mp 5 . . . 4 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))))
21 r2al 2923 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2217, 20, 213bitr4i 291 . . 3 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2322anbi2i 726 . 2 ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
241, 23bitr4i 266 1 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∨ w3o 1030  ∀wal 1473   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   I cid 4948   Po wpo 4957   Or wor 4958   × cxp 5036  ◡ccnv 5037  Rel wrel 5043 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  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-br 4584  df-opab 4644  df-id 4953  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator