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Theorem cnvpom 4887
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 2458 . . . . . . 7 (∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
2 ralidm 3348 . . . . . . . . 9 (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴 ¬ 𝑤𝑅𝑤)
3 r19.3rmv 3339 . . . . . . . . . 10 (∃𝑥 𝑥𝐴 → (¬ 𝑤𝑅𝑤 ↔ ∀𝑧𝐴 ¬ 𝑤𝑅𝑤))
43ralbidv 2343 . . . . . . . . 9 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤))
52, 4syl5rbb 186 . . . . . . . 8 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ↔ ∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤))
65anbi1d 446 . . . . . . 7 (∃𝑥 𝑥𝐴 → ((∀𝑤𝐴𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
71, 6syl5bb 185 . . . . . 6 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
8 r19.26 2458 . . . . . . 7 (∀𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
98ralbii 2347 . . . . . 6 (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴 (∀𝑧𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
10 r19.26 2458 . . . . . 6 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ (∀𝑤𝐴𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑤𝐴𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
117, 9, 103bitr4g 216 . . . . 5 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))))
12 r19.26 2458 . . . . . . . 8 (∀𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ (∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ∧ ∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
13 vex 2577 . . . . . . . . . . . . 13 𝑧 ∈ V
1413, 13brcnv 4545 . . . . . . . . . . . 12 (𝑧𝑅𝑧𝑧𝑅𝑧)
15 id 19 . . . . . . . . . . . . 13 (𝑧 = 𝑤𝑧 = 𝑤)
1615, 15breq12d 3804 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝑧𝑅𝑧𝑤𝑅𝑤))
1714, 16syl5bb 185 . . . . . . . . . . 11 (𝑧 = 𝑤 → (𝑧𝑅𝑧𝑤𝑅𝑤))
1817notbid 602 . . . . . . . . . 10 (𝑧 = 𝑤 → (¬ 𝑧𝑅𝑧 ↔ ¬ 𝑤𝑅𝑤))
1918cbvralv 2550 . . . . . . . . 9 (∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ↔ ∀𝑤𝐴 ¬ 𝑤𝑅𝑤)
20 vex 2577 . . . . . . . . . . . . 13 𝑦 ∈ V
2113, 20brcnv 4545 . . . . . . . . . . . 12 (𝑧𝑅𝑦𝑦𝑅𝑧)
22 vex 2577 . . . . . . . . . . . . 13 𝑤 ∈ V
2320, 22brcnv 4545 . . . . . . . . . . . 12 (𝑦𝑅𝑤𝑤𝑅𝑦)
2421, 23anbi12ci 442 . . . . . . . . . . 11 ((𝑧𝑅𝑦𝑦𝑅𝑤) ↔ (𝑤𝑅𝑦𝑦𝑅𝑧))
2513, 22brcnv 4545 . . . . . . . . . . 11 (𝑧𝑅𝑤𝑤𝑅𝑧)
2624, 25imbi12i 232 . . . . . . . . . 10 (((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤) ↔ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
2726ralbii 2347 . . . . . . . . 9 (∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤) ↔ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧))
2819, 27anbi12i 441 . . . . . . . 8 ((∀𝑧𝐴 ¬ 𝑧𝑅𝑧 ∧ ∀𝑧𝐴 ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
2912, 28bitr2i 178 . . . . . . 7 ((∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3029ralbii 2347 . . . . . 6 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑤𝐴𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
31 ralcom 2490 . . . . . 6 (∀𝑤𝐴𝑧𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3230, 31bitri 177 . . . . 5 (∀𝑤𝐴 (∀𝑤𝐴 ¬ 𝑤𝑅𝑤 ∧ ∀𝑧𝐴 ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3311, 32syl6bb 189 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
3433ralbidv 2343 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑦𝐴𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦𝐴𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
35 ralcom 2490 . . 3 (∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑦𝐴𝑤𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
36 ralcom 2490 . . 3 (∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)) ↔ ∀𝑦𝐴𝑧𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
3734, 35, 363bitr4g 216 . 2 (∃𝑥 𝑥𝐴 → (∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)) ↔ ∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤))))
38 df-po 4060 . 2 (𝑅 Po 𝐴 ↔ ∀𝑤𝐴𝑦𝐴𝑧𝐴𝑤𝑅𝑤 ∧ ((𝑤𝑅𝑦𝑦𝑅𝑧) → 𝑤𝑅𝑧)))
39 df-po 4060 . 2 (𝑅 Po 𝐴 ↔ ∀𝑧𝐴𝑦𝐴𝑤𝐴𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑤) → 𝑧𝑅𝑤)))
4037, 38, 393bitr4g 216 1 (∃𝑥 𝑥𝐴 → (𝑅 Po 𝐴𝑅 Po 𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  wex 1397  wcel 1409  wral 2323   class class class wbr 3791   Po wpo 4058  ccnv 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-po 4060  df-cnv 4380
This theorem is referenced by:  cnvsom  4888
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