Theorem cnvpom 4880

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

Assertion

Ref	Expression
cnvpom	⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem cnvpom

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

Step	Hyp	Ref	Expression
1		r19.26 2485	. . . . . . 7 ⊢ (∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
2		ralidm 3341	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤)
3		r19.3rmv 3332	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (¬ 𝑤𝑅𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
4	3	ralbidv 2368	. . . . . . . . 9 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
5	2, 4	syl5rbb 191	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
6	5	anbi1d 452	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ((∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
7	1, 6	syl5bb 190	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
8		r19.26 2485	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
9	8	ralbii 2372	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
10		r19.26 2485	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
11	7, 9, 10	3bitr4g 221	. . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
12		r19.26 2485	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
13		vex 2604	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
14	13, 13	brcnv 4536	. . . . . . . . . . . 12 ⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧)
15		id 19	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
16	15, 15	breq12d 3798	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑧 ↔ 𝑤𝑅𝑤))
17	14, 16	syl5bb 190	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧◡𝑅𝑧 ↔ 𝑤𝑅𝑤))
18	17	notbid 624	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑤𝑅𝑤))
19	18	cbvralv 2577	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤)
20		vex 2604	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	13, 20	brcnv 4536	. . . . . . . . . . . 12 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
22		vex 2604	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
23	20, 22	brcnv 4536	. . . . . . . . . . . 12 ⊢ (𝑦◡𝑅𝑤 ↔ 𝑤𝑅𝑦)
24	21, 23	anbi12ci 448	. . . . . . . . . . 11 ⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) ↔ (𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧))
25	13, 22	brcnv 4536	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑤 ↔ 𝑤𝑅𝑧)
26	24, 25	imbi12i 237	. . . . . . . . . 10 ⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
27	26	ralbii 2372	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
28	19, 27	anbi12i 447	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ∧ ∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
29	12, 28	bitr2i 183	. . . . . . 7 ⊢ ((∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
30	29	ralbii 2372	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
31		ralcom 2517	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
32	30, 31	bitri 182	. . . . 5 ⊢ (∀𝑤 ∈ 𝐴 (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
33	11, 32	syl6bb 194	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))))
34	33	ralbidv 2368	. . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))))
35		ralcom 2517	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
36		ralcom 2517	. . 3 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
37	34, 35, 36	3bitr4g 221	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤))))
38		df-po 4051	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
39		df-po 4051	. 2 ⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
40	37, 38, 39	3bitr4g 221	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∃wex 1421   ∈ wcel 1433  ∀wral 2348   class class class wbr 3785   Po wpo 4049  ^◡ccnv 4362

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-po 4051  df-cnv 4371

This theorem is referenced by:  cnvsom  4881