Theorem dfso2 34725

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 5590	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
2		opelxp 5713	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		brun 5200	. . . . . . . . . 10 ⊢ (𝑥( I ∪ ◡𝑅)𝑦 ↔ (𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦))
4		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
5	4	ideq 5853	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
6		vex 3479	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5883	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 7	orbi12i 914	. . . . . . . . . 10 ⊢ ((𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	3, 8	bitr2i 276	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥( I ∪ ◡𝑅)𝑦)
10	9	orbi2i 912	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
11		3orass 1091	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
12		brun 5200	. . . . . . . 8 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
13	10, 11, 12	3bitr4i 303	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦)
14		df-br 5150	. . . . . . 7 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)))
15	13, 14	bitr2i 276	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	2, 15	imbi12i 351	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
17	16	2albii 1823	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
18		relxp 5695	. . . . 5 ⊢ Rel (𝐴 × 𝐴)
19		ssrel 5783	. . . . 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 3195	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	17, 20, 21	3bitr4i 303	. . 3 ⊢ ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	22	anbi2i 624	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
24	1, 23	bitr4i 278	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∨ w3o 1087  ∀wal 1540   ∈ wcel 2107  ∀wral 3062   ∪ cun 3947   ⊆ wss 3949  ⟨cop 4635   class class class wbr 5149   I cid 5574   Po wpo 5587   Or wor 5588   × cxp 5675  ^◡ccnv 5676  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685

This theorem is referenced by: (None)