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Theorem pocl 4121
Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)
Assertion
Ref Expression
pocl (𝑅 Po 𝐴 → ((𝐵𝐴𝐶𝐴𝐷𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Proof of Theorem pocl
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . . 7 (𝑥 = 𝐵𝑥 = 𝐵)
21, 1breq12d 3850 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑥𝐵𝑅𝐵))
32notbid 627 . . . . 5 (𝑥 = 𝐵 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
4 breq1 3840 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
54anbi1d 453 . . . . . 6 (𝑥 = 𝐵 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝐵𝑅𝑦𝑦𝑅𝑧)))
6 breq1 3840 . . . . . 6 (𝑥 = 𝐵 → (𝑥𝑅𝑧𝐵𝑅𝑧))
75, 6imbi12d 232 . . . . 5 (𝑥 = 𝐵 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧)))
83, 7anbi12d 457 . . . 4 (𝑥 = 𝐵 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧))))
98imbi2d 228 . . 3 (𝑥 = 𝐵 → ((𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧)))))
10 breq2 3841 . . . . . . 7 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
11 breq1 3840 . . . . . . 7 (𝑦 = 𝐶 → (𝑦𝑅𝑧𝐶𝑅𝑧))
1210, 11anbi12d 457 . . . . . 6 (𝑦 = 𝐶 → ((𝐵𝑅𝑦𝑦𝑅𝑧) ↔ (𝐵𝑅𝐶𝐶𝑅𝑧)))
1312imbi1d 229 . . . . 5 (𝑦 = 𝐶 → (((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧)))
1413anbi2d 452 . . . 4 (𝑦 = 𝐶 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧))))
1514imbi2d 228 . . 3 (𝑦 = 𝐶 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦𝑦𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧)))))
16 breq2 3841 . . . . . . 7 (𝑧 = 𝐷 → (𝐶𝑅𝑧𝐶𝑅𝐷))
1716anbi2d 452 . . . . . 6 (𝑧 = 𝐷 → ((𝐵𝑅𝐶𝐶𝑅𝑧) ↔ (𝐵𝑅𝐶𝐶𝑅𝐷)))
18 breq2 3841 . . . . . 6 (𝑧 = 𝐷 → (𝐵𝑅𝑧𝐵𝑅𝐷))
1917, 18imbi12d 232 . . . . 5 (𝑧 = 𝐷 → (((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
2019anbi2d 452 . . . 4 (𝑧 = 𝐷 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
2120imbi2d 228 . . 3 (𝑧 = 𝐷 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷)))))
22 df-po 4114 . . . . . . . 8 (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
23 r3al 2420 . . . . . . . 8 (∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2422, 23bitri 182 . . . . . . 7 (𝑅 Po 𝐴 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2524biimpi 118 . . . . . 6 (𝑅 Po 𝐴 → ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
262519.21bbi 1496 . . . . 5 (𝑅 Po 𝐴 → ∀𝑧((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
272619.21bi 1495 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝐴𝑦𝐴𝑧𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
2827com12 30 . . 3 ((𝑥𝐴𝑦𝐴𝑧𝐴) → (𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
299, 15, 21, 28vtocl3ga 2689 . 2 ((𝐵𝐴𝐶𝐴𝐷𝐴) → (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
3029com12 30 1 (𝑅 Po 𝐴 → ((𝐵𝐴𝐶𝐴𝐷𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 924  wal 1287   = wceq 1289  wcel 1438  wral 2359   class class class wbr 3837   Po wpo 4112
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-po 4114
This theorem is referenced by:  poirr  4125  potr  4126
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