Theorem cnvpo 6137

Description: The converse of a partial order relation is a partial order relation. (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 3170	. . . . . . 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 3497	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
3	2, 2	brcnv 5752	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧)
4		id 22	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
5	4, 4	breq12d 5078	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑧 ↔ 𝑥𝑅𝑥))
6	3, 5	syl5bb 285	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧◡𝑅𝑧 ↔ 𝑥𝑅𝑥))
7	6	notbid 320	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑥𝑅𝑥))
8	7	cbvralvw 3449	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
9		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
10	2, 9	brcnv 5752	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
11		vex 3497	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12	9, 11	brcnv 5752	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
13	10, 12	anbi12ci 629	. . . . . . . . . 10 ⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧))
14	2, 11	brcnv 5752	. . . . . . . . . 10 ⊢ (𝑧◡𝑅𝑥 ↔ 𝑥𝑅𝑧)
15	13, 14	imbi12i 353	. . . . . . . . 9 ⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
16	15	ralbii 3165	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
17	8, 16	anbi12i 628	. . . . . . 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 278	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
19	18	ralbii 3165	. . . . 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 3170	. . . . . . 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 4454	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
22		rzal 4452	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
23		rzal 4452	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
24	22, 23	2thd 267	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
25		r19.3rzv 4443	. . . . . . . . . . 11 ⊢ (𝐴 ≠ ∅ → (¬ 𝑥𝑅𝑥 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3197	. . . . . . . . . 10 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥))
27	24, 26	pm2.61ine 3100	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
28	21, 27	bitr2i 278	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥)
29	28	anbi1i 625	. . . . . . 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 277	. . . . . 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 3170	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
32	31	ralbii 3165	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
33		r19.26 3170	. . . . . 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 305	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ 𝐴 (∀𝑥 ∈ 𝐴 ¬ 𝑥𝑅𝑥 ∧ ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
35		ralcom 3354	. . . . 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 305	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
37	36	ralbii 3165	. . 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 3354	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
39		ralcom 3354	. . 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 305	. 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 5473	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
42		df-po 5473	. 2 ⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑥) → 𝑧◡𝑅𝑥)))
43	40, 41, 42	3bitr4i 305	1 ⊢ (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ≠ wne 3016  ∀wral 3138  ∅c0 4290   class class class wbr 5065   Po wpo 5471  ^◡ccnv 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-po 5473  df-cnv 5562

This theorem is referenced by:  cnvso  6138  fimax2g  8763  fin23lem40  9772  isfin1-3  9807