Theorem cnvpo 6318

Description: The converse of a partial order is a partial order. (Contributed by NM, 15-Jun-2005.)

Assertion

Ref	Expression
cnvpo	⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)

Proof of Theorem cnvpo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 3117	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
2		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
3	2, 2	brcnv 5907	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧)
4		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	4, 4	breq12d 5179	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑧 ↔ 𝑥𝑅𝑥))
6	3, 5	bitrid 283	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧◡𝑅𝑧 ↔ 𝑥𝑅𝑥))
7	6	notbid 318	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑥𝑅𝑥))
8	7	cbvralvw 3243	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
9		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
10	2, 9	brcnv 5907	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
11		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12	9, 11	brcnv 5907	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
13	10, 12	anbi12ci 628	. . . . . . . . . 10 ⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧))
14	2, 11	brcnv 5907	. . . . . . . . . 10 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
15	13, 14	imbi12i 350	. . . . . . . . 9 ⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
16	15	ralbii 3099	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
17	8, 16	anbi12i 627	. . . . . . 7 ⊢ ((∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)) ↔ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
18	1, 17	bitr2i 276	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
19	18	ralbii 3099	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
20		r19.26 3117	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
21		ralidm 4535	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
22		rzal 4532	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
23		rzal 4532	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
24	22, 23	2thd 265	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
25		r19.3rzv 4522	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (¬ 𝑥𝑅𝑥 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3184	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
27	24, 26	pm2.61ine 3031	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
28	21, 27	bitr2i 276	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
29	28	anbi1i 623	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
30	20, 29	bitri 275	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
31		r19.26 3117	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32	31	ralbii 3099	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
33		r19.26 3117	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
34	30, 32, 33	3bitr4i 303	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
35		ralcom 3295	. . . . 5 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
36	19, 34, 35	3bitr4i 303	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
37	36	ralbii 3099	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
38		ralcom 3295	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
39		ralcom 3295	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
40	37, 38, 39	3bitr4i 303	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
41		df-po 5607	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
42		df-po 5607	. 2 ⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
43	40, 41, 42	3bitr4i 303	1 ⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ≠ wne 2946  ∀wral 3067  ∅c0 4352   class class class wbr 5166   Po wpo 5605  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-po 5607  df-cnv 5708

This theorem is referenced by:  cnvso  6319  fimax2g  9350  fin23lem40  10420  isfin1-3  10455