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Theorem dfso2 31352
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 4996 . 2 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2 opelxp 5106 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥𝐴𝑦𝐴))
3 brun 4663 . . . . . . . . . 10 (𝑥( I ∪ 𝑅)𝑦 ↔ (𝑥 I 𝑦𝑥𝑅𝑦))
4 vex 3189 . . . . . . . . . . . 12 𝑦 ∈ V
54ideq 5234 . . . . . . . . . . 11 (𝑥 I 𝑦𝑥 = 𝑦)
6 vex 3189 . . . . . . . . . . . 12 𝑥 ∈ V
76, 4brcnv 5265 . . . . . . . . . . 11 (𝑥𝑅𝑦𝑦𝑅𝑥)
85, 7orbi12i 543 . . . . . . . . . 10 ((𝑥 I 𝑦𝑥𝑅𝑦) ↔ (𝑥 = 𝑦𝑦𝑅𝑥))
93, 8bitr2i 265 . . . . . . . . 9 ((𝑥 = 𝑦𝑦𝑅𝑥) ↔ 𝑥( I ∪ 𝑅)𝑦)
109orbi2i 541 . . . . . . . 8 ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦𝑥( I ∪ 𝑅)𝑦))
11 3orass 1039 . . . . . . . 8 ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦𝑦𝑅𝑥)))
12 brun 4663 . . . . . . . 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 4614 . . . . . . 7 (𝑥(𝑅 ∪ ( I ∪ 𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)))
1513, 14bitr2i 265 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅)) ↔ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
162, 15imbi12i 340 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
17162albii 1745 . . . 4 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ 𝑅))) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 relxp 5188 . . . . 5 Rel (𝐴 × 𝐴)
19 ssrel 5168 . . . . 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 2934 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2217, 20, 213bitr4i 292 . . 3 ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
2322anbi2i 729 . 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 383  wa 384  w3o 1035  wal 1478  wcel 1987  wral 2907  cun 3553  wss 3555  cop 4154   class class class wbr 4613   I cid 4984   Po wpo 4993   Or wor 4994   × cxp 5072  ccnv 5073  Rel wrel 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082
This theorem is referenced by: (None)
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