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Theorem elpotr 33757
Description: A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
Assertion
Ref Expression
elpotr (∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)
Distinct variable group:   𝑧,𝐴

Proof of Theorem elpotr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alral 3080 . . . . . 6 (∀𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
21alimi 1814 . . . . 5 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
3 alral 3080 . . . . 5 (∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
42, 3syl 17 . . . 4 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
54ralimi 3087 . . 3 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
6 ralcom 3166 . . . 4 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
7 ralcom 3166 . . . . 5 (∀𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87ralbii 3092 . . . 4 (∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
96, 8bitri 274 . . 3 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
105, 9sylib 217 . 2 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
11 dftr2 5193 . . 3 (Tr 𝑧 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1211ralbii 3092 . 2 (∀𝑧𝐴 Tr 𝑧 ↔ ∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
13 df-po 5503 . . 3 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
14 epel 5498 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
15 epel 5498 . . . . . . . 8 (𝑦 E 𝑧𝑦𝑧)
1614, 15anbi12i 627 . . . . . . 7 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
17 epel 5498 . . . . . . 7 (𝑥 E 𝑧𝑥𝑧)
1816, 17imbi12i 351 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
19 elirrv 9355 . . . . . . . 8 ¬ 𝑥𝑥
20 epel 5498 . . . . . . . 8 (𝑥 E 𝑥𝑥𝑥)
2119, 20mtbir 323 . . . . . . 7 ¬ 𝑥 E 𝑥
2221biantrur 531 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2318, 22bitr3i 276 . . . . 5 (((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2423ralbii 3092 . . . 4 (∀𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
25242ralbii 3093 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2613, 25bitr4i 277 . 2 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
2710, 12, 263imtr4i 292 1 (∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1537  wral 3064   class class class wbr 5074  Tr wtr 5191   E cep 5494   Po wpo 5501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-reg 9351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503
This theorem is referenced by: (None)
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