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Theorem dfso2 33440
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 5469 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2 opelxp 5587 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
3 brun 5104 . . . . . . . . . 10 (𝑥( I ∪ 𝑅)𝑦 ↔ (𝑥 I 𝑦𝑥𝑅𝑦))
4 vex 3412 . . . . . . . . . . . 12 𝑦 ∈ V
54ideq 5721 . . . . . . . . . . 11 (𝑥 I 𝑦𝑥 = 𝑦)
6 vex 3412 . . . . . . . . . . . 12 𝑥 ∈ V
76, 4brcnv 5751 . . . . . . . . . . 11 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 7orbi12i 915 . . . . . . . . . 10 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑦𝑅𝑥))
93, 8bitr2i 279 . . . . . . . . 9 ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥( I ∪ 𝑅)𝑦)
109orbi2i 913 . . . . . . . 8 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
11 3orass 1092 . . . . . . . 8 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
12 brun 5104 . . . . . . . 8 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
1310, 11, 123bitr4i 306 . . . . . . 7 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦)
14 df-br 5054 . . . . . . 7 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))
1513, 14bitr2i 279 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
162, 15imbi12i 354 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
17162albii 1828 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 relxp 5569 . . . . 5 Rel (𝐴 × 𝐴)
19 ssrel 5654 . . . . 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 3122 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2217, 20, 213bitr4i 306 . . 3 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2322anbi2i 626 . 2 ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
241, 23bitr4i 281 1 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847  w3o 1088  wal 1541  wcel 2110  wral 3061  cun 3864  wss 3866  cop 4547   class class class wbr 5053   I cid 5454   Po wpo 5466   Or wor 5467   × cxp 5549  ccnv 5550  Rel wrel 5556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-so 5469  df-xp 5557  df-rel 5558  df-cnv 5559
This theorem is referenced by: (None)
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