Theorem dfso2 32600

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.

Step	Hyp	Ref	Expression
1		df-so 5370	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
2		opelxp 5486	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		brun 5019	. . . . . . . . . 10 ⊢ (𝑥( I ∪ ◡𝑅)𝑦 ↔ (𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦))
4		vex 3443	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
5	4	ideq 5616	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
6		vex 3443	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5646	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 7	orbi12i 909	. . . . . . . . . 10 ⊢ ((𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	3, 8	bitr2i 277	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥( I ∪ ◡𝑅)𝑦)
10	9	orbi2i 907	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
11		3orass 1083	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
12		brun 5019	. . . . . . . 8 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
13	10, 11, 12	3bitr4i 304	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦)
14		df-br 4969	. . . . . . 7 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)))
15	13, 14	bitr2i 277	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	2, 15	imbi12i 352	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
17	16	2albii 1806	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
18		relxp 5468	. . . . 5 ⊢ Rel (𝐴 × 𝐴)
19		ssrel 5550	. . . . 5 ⊢ (Rel (𝐴 × 𝐴) → ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)))))
20	18, 19	ax-mp 5	. . . 4 ⊢ ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))))
21		r2al 3170	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	17, 20, 21	3bitr4i 304	. . 3 ⊢ ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	22	anbi2i 622	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
24	1, 23	bitr4i 279	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∨ w3o 1079  ∀wal 1523   ∈ wcel 2083  ∀wral 3107   ∪ cun 3863   ⊆ wss 3865  ⟨cop 4484   class class class wbr 4968   I cid 5354   Po wpo 5367   Or wor 5368   × cxp 5448  ^◡ccnv 5449  Rel wrel 5455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-id 5355  df-so 5370  df-xp 5456  df-rel 5457  df-cnv 5458

This theorem is referenced by: (None)