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Theorem cnvpom 4910
Description: The converse of a partial order relation is a partial order relation. (Contributed by NM, 15-Jun-2005.)
Assertion
Ref Expression
cnvpom (∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 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.
StepHypRef Expression
1 r19.26 2490 . . . . . . 7 (∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
2 ralidm 3358 . . . . . . . . 9 (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴 ¬ 𝑤𝑅𝑤)
3 r19.3rmv 3348 . . . . . . . . . 10 (∃𝑥 𝑥𝐴 → (¬ 𝑤𝑅𝑤 ↔ ∀𝑧𝐴 ¬ 𝑤𝑅𝑤))
43ralbidv 2373 . . . . . . . . 9 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤))
52, 4syl5rbb 191 . . . . . . . 8 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤))
65anbi1d 453 . . . . . . 7 (∃𝑥 𝑥𝐴 → ((∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
71, 6syl5bb 190 . . . . . 6 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
8 r19.26 2490 . . . . . . 7 (∀𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
98ralbii 2377 . . . . . 6 (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
10 r19.26 2490 . . . . . 6 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
117, 9, 103bitr4g 221 . . . . 5 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
12 r19.26 2490 . . . . . . . 8 (∀𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ (∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ∧ ∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
13 vex 2613 . . . . . . . . . . . . 13 𝑧 ∈ V
1413, 13brcnv 4566 . . . . . . . . . . . 12 (𝑧𝑅𝑧𝑧𝑅𝑧)
15 id 19 . . . . . . . . . . . . 13 (𝑧 = 𝑤𝑧 = 𝑤)
1615, 15breq12d 3818 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑅𝑧𝑤𝑅𝑤))
1714, 16syl5bb 190 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝑅𝑧𝑤𝑅𝑤))
1817notbid 625 . . . . . . . . . 10 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑧 ↔ ¬ 𝑤𝑅𝑤))
1918cbvralv 2582 . . . . . . . . 9 (∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ↔ ∀𝑤𝐴 ¬ 𝑤𝑅𝑤)
20 vex 2613 . . . . . . . . . . . . 13 𝑦 ∈ V
2113, 20brcnv 4566 . . . . . . . . . . . 12 (𝑧𝑅𝑦𝑦𝑅𝑧)
22 vex 2613 . . . . . . . . . . . . 13 𝑤 ∈ V
2320, 22brcnv 4566 . . . . . . . . . . . 12 (𝑦𝑅𝑤𝑤𝑅𝑦)
2421, 23anbi12ci 449 . . . . . . . . . . 11 ((𝑧𝑅𝑦𝑦𝑅𝑤) ↔ (𝑤𝑅𝑦𝑦𝑅𝑧))
2513, 22brcnv 4566 . . . . . . . . . . 11 (𝑧𝑅𝑤𝑤𝑅𝑧)
2624, 25imbi12i 237 . . . . . . . . . 10 (((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤) ↔ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
2726ralbii 2377 . . . . . . . . 9 (∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤) ↔ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
2819, 27anbi12i 448 . . . . . . . 8 ((∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ∧ ∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
2912, 28bitr2i 183 . . . . . . 7 ((∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3029ralbii 2377 . . . . . 6 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
31 ralcom 2522 . . . . . 6 (∀𝑤𝐴𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3230, 31bitri 182 . . . . 5 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3311, 32syl6bb 194 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
3433ralbidv 2373 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑦𝐴𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦𝐴𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
35 ralcom 2522 . . 3 (∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦𝐴𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
36 ralcom 2522 . . 3 (∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ ∀𝑦𝐴𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3734, 35, 363bitr4g 221 . 2 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
38 df-po 4079 . 2 (𝑅 Po 𝐴 ↔ ∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
39 df-po 4079 . 2 (𝑅 Po 𝐴 ↔ ∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
4037, 38, 393bitr4g 221 1 (∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wex 1422  wcel 1434  wral 2353   class class class wbr 3805   Po wpo 4077  ccnv 4390
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-po 4079  df-cnv 4399
This theorem is referenced by:  cnvsom  4911
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