Theorem dfso2 36697

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 5564	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
2		opelxp 5691	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴))
3		brun 5159	. . . . . . . . . 10 ⊢ (𝑥( I ∪ ◡𝑅)𝑦 ↔ (𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦))
4		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
5	4	ideq 5835	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
6		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5866	. . . . . . . . . . 11 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
8	5, 7	orbi12i 928	. . . . . . . . . 10 ⊢ ((𝑥 I 𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
9	3, 8	bitr2i 279	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥( I ∪ ◡𝑅)𝑦)
10	9	orbi2i 926	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
11		3orass 1106	. . . . . . . 8 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
12		brun 5159	. . . . . . . 8 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥( I ∪ ◡𝑅)𝑦))
13	10, 11, 12	3bitr4i 306	. . . . . . 7 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ 𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦)
14		df-br 5107	. . . . . . 7 ⊢ (𝑥(𝑅 ∪ ( I ∪ ◡𝑅))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)))
15	13, 14	bitr2i 279	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	2, 15	imbi12i 353	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
17	16	2albii 1853	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
18		relxp 5673	. . . . 5 ⊢ Rel (𝐴 × 𝐴)
19		ssrel 5764	. . . . 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 3202	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	17, 20, 21	3bitr4i 306	. . 3 ⊢ ((𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	22	anbi2i 635	. 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))) ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
24	1, 23	bitr4i 281	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102  ∀wal 1568   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900   ∪ cun 4152  ⟨cop 4838   class class class wbr 5106   I cid 5549   Po wpo 5561   Or wor 5562   × cxp 5653  ^◡ccnv 5654  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-so 5564  df-xp 5661  df-rel 5662  df-cnv 5663

This theorem is used by: (None)