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Theorem dfso2 35735
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 5598 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2 opelxp 5725 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
3 brun 5199 . . . . . . . . . 10 (𝑥( I ∪ 𝑅)𝑦 ↔ (𝑥 I 𝑦𝑥𝑅𝑦))
4 vex 3482 . . . . . . . . . . . 12 𝑦 ∈ V
54ideq 5866 . . . . . . . . . . 11 (𝑥 I 𝑦𝑥 = 𝑦)
6 vex 3482 . . . . . . . . . . . 12 𝑥 ∈ V
76, 4brcnv 5896 . . . . . . . . . . 11 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 7orbi12i 914 . . . . . . . . . 10 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑦𝑅𝑥))
93, 8bitr2i 276 . . . . . . . . 9 ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥( I ∪ 𝑅)𝑦)
109orbi2i 912 . . . . . . . 8 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
11 3orass 1089 . . . . . . . 8 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
12 brun 5199 . . . . . . . 8 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
1310, 11, 123bitr4i 303 . . . . . . 7 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦)
14 df-br 5149 . . . . . . 7 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))
1513, 14bitr2i 276 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
162, 15imbi12i 350 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
17162albii 1817 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 relxp 5707 . . . . 5 Rel (𝐴 × 𝐴)
19 ssrel 5795 . . . . 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 3193 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2217, 20, 213bitr4i 303 . . 3 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2322anbi2i 623 . 2 ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
241, 23bitr4i 278 1 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3o 1085  wal 1535  wcel 2106  wral 3059  cun 3961  wss 3963  cop 4637   class class class wbr 5148   I cid 5582   Po wpo 5595   Or wor 5596   × cxp 5687  ccnv 5688  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697
This theorem is referenced by: (None)
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