Theorem cnvpo 6288

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 3124	. . . . . . 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 3458	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
3	2, 2	brcnv 5867	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧)
4		id 23	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	4, 4	breq12d 5121	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑧 ↔ 𝑥𝑅𝑥))
6	3, 5	bitrid 286	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧◡𝑅𝑧 ↔ 𝑥𝑅𝑥))
7	6	notbid 321	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑥𝑅𝑥))
8	7	cbvralvw 3242	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
9		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
10	2, 9	brcnv 5867	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
11		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12	9, 11	brcnv 5867	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
13	10, 12	anbi12ci 640	. . . . . . . . . 10 ⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧))
14	2, 11	brcnv 5867	. . . . . . . . . 10 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
15	13, 14	imbi12i 353	. . . . . . . . 9 ⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
16	15	ralbii 3110	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
17	8, 16	anbi12i 639	. . . . . . 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 279	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
19	18	ralbii 3110	. . . . 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 3124	. . . . . . 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 4477	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
22		rzal 4454	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
23		rzal 4454	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
24	22, 23	2thd 268	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
25		r19.3rzv 4463	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (¬ 𝑥𝑅𝑥 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3187	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
27	24, 26	pm2.61ine 3040	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
28	21, 27	bitr2i 279	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
29	28	anbi1i 635	. . . . . . 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 278	. . . . . 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 3124	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32	31	ralbii 3110	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
33		r19.26 3124	. . . . . 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 306	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
35		ralcom 3292	. . . . 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 306	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
37	36	ralbii 3110	. . 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 3292	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
39		ralcom 3292	. . 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 306	. 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 5568	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
42		df-po 5568	. 2 ⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
43	40, 41, 42	3bitr4i 306	1 ⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ≠ wne 2957  ∀wral 3078  ∅c0 4285   class class class wbr 5108   Po wpo 5566  ^◡ccnv 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-po 5568  df-cnv 5668

This theorem is used by:  cnvso  6289  fimax2g  9244  fin23lem40  10341  isfin1-3  10376