Theorem dfso2 31644

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 5036	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
2		opelxp 5146	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		brun 4703	. . . . . . . . . 10 ⊢ (𝑥( I ∪ ◡𝑅)𝑦 ↔ (𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦))
4		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
5	4	ideq 5274	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
6		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5305	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 7	orbi12i 543	. . . . . . . . . 10 ⊢ ((𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	3, 8	bitr2i 265	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥( I ∪ ◡𝑅)𝑦)
10	9	orbi2i 541	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
11		3orass 1040	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
12		brun 4703	. . . . . . . 8 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
13	10, 11, 12	3bitr4i 292	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦)
14		df-br 4654	. . . . . . 7 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)))
15	13, 14	bitr2i 265	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	2, 15	imbi12i 340	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
17	16	2albii 1748	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
18		relxp 5227	. . . . 5 ⊢ Rel (𝐴 × 𝐴)
19		ssrel 5207	. . . . 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 2939	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	17, 20, 21	3bitr4i 292	. . 3 ⊢ ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	22	anbi2i 730	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
24	1, 23	bitr4i 267	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036  ∀wal 1481   ∈ wcel 1990  ∀wral 2912   ∪ cun 3572   ⊆ wss 3574  ⟨cop 4183   class class class wbr 4653   I cid 5023   Po wpo 5033   Or wor 5034   × cxp 5112  ^◡ccnv 5113  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)