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Theorem dfso2 31973
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 5189 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2 opelxp 5304 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
3 brun 4856 . . . . . . . . . 10 (𝑥( I ∪ 𝑅)𝑦 ↔ (𝑥 I 𝑦𝑥𝑅𝑦))
4 vex 3344 . . . . . . . . . . . 12 𝑦 ∈ V
54ideq 5431 . . . . . . . . . . 11 (𝑥 I 𝑦𝑥 = 𝑦)
6 vex 3344 . . . . . . . . . . . 12 𝑥 ∈ V
76, 4brcnv 5461 . . . . . . . . . . 11 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 7orbi12i 544 . . . . . . . . . 10 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑦𝑅𝑥))
93, 8bitr2i 265 . . . . . . . . 9 ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥( I ∪ 𝑅)𝑦)
109orbi2i 542 . . . . . . . 8 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
11 3orass 1075 . . . . . . . 8 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
12 brun 4856 . . . . . . . 8 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
1310, 11, 123bitr4i 292 . . . . . . 7 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦)
14 df-br 4806 . . . . . . 7 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))
1513, 14bitr2i 265 . . . . . 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 1897 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 relxp 5284 . . . . 5 Rel (𝐴 × 𝐴)
19 ssrel 5365 . . . . 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 3078 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2217, 20, 213bitr4i 292 . . 3 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2322anbi2i 732 . 2 ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
241, 23bitr4i 267 1 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3o 1071  wal 1630  wcel 2140  wral 3051  cun 3714  wss 3716  cop 4328   class class class wbr 4805   I cid 5174   Po wpo 5186   Or wor 5187   × cxp 5265  ccnv 5266  Rel wrel 5272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-id 5175  df-so 5189  df-xp 5273  df-rel 5274  df-cnv 5275
This theorem is referenced by: (None)
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