Theorem cnvpom 5039

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 2532	. . . . . . 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 3429	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤)
3		r19.3rmv 3419	. . . . . . . . . 10 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (¬ 𝑤𝑅𝑤 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
4	3	ralbidv 2411	. . . . . . . . 9 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
5	2, 4	syl5rbb 192	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤))
6	5	anbi1d 458	. . . . . . 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 191	. . . . . 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 2532	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
9	8	ralbii 2415	. . . . . 6 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤 ∈ 𝐴 (∀𝑧 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
10		r19.26 2532	. . . . . 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 222	. . . . 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 2532	. . . . . . . 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 2660	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
14	13, 13	brcnv 4682	. . . . . . . . . . . 12 ⊢ (𝑧◡𝑅𝑧 ↔ 𝑧𝑅𝑧)
15		id 19	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → 𝑧 = 𝑤)
16	15, 15	breq12d 3908	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑧 ↔ 𝑤𝑅𝑤))
17	14, 16	syl5bb 191	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝑧◡𝑅𝑧 ↔ 𝑤𝑅𝑤))
18	17	notbid 639	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (¬ 𝑧◡𝑅𝑧 ↔ ¬ 𝑤𝑅𝑤))
19	18	cbvralv 2628	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ¬ 𝑧◡𝑅𝑧 ↔ ∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤)
20		vex 2660	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
21	13, 20	brcnv 4682	. . . . . . . . . . . 12 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
22		vex 2660	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
23	20, 22	brcnv 4682	. . . . . . . . . . . 12 ⊢ (𝑦◡𝑅𝑤 ↔ 𝑤𝑅𝑦)
24	21, 23	anbi12ci 454	. . . . . . . . . . 11 ⊢ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) ↔ (𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧))
25	13, 22	brcnv 4682	. . . . . . . . . . 11 ⊢ (𝑧◡𝑅𝑤 ↔ 𝑤𝑅𝑧)
26	24, 25	imbi12i 238	. . . . . . . . . 10 ⊢ (((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
27	26	ralbii 2415	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤) ↔ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧))
28	19, 27	anbi12i 453	. . . . . . . 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 184	. . . . . . 7 ⊢ ((∀𝑤 ∈ 𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧 ∈ 𝐴 ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
30	29	ralbii 2415	. . . . . 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 2568	. . . . . 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 183	. . . . 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 195	. . . 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 2411	. . 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 2568	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
36		ralcom 2568	. . 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 222	. 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 4178	. 2 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
39		df-po 4178	. 2 ⊢ (◡𝑅 Po 𝐴 ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (¬ 𝑧◡𝑅𝑧 ∧ ((𝑧◡𝑅𝑦 ∧ 𝑦◡𝑅𝑤) → 𝑧◡𝑅𝑤)))
40	37, 38, 39	3bitr4g 222	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝑅 Po 𝐴 ↔ ◡𝑅 Po 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  ∃wex 1451   ∈ wcel 1463  ∀wral 2390   class class class wbr 3895   Po wpo 4176  ^◡ccnv 4498

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-po 4178  df-cnv 4507

This theorem is referenced by:  cnvsom  5040