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Theorem elpotr 31810
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 2957 . . . . . 6 (∀𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
21alimi 1779 . . . . 5 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
3 alral 2957 . . . . 5 (∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
42, 3syl 17 . . . 4 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
54ralimi 2981 . . 3 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
6 ralcom 3127 . . . 4 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
7 ralcom 3127 . . . . 5 (∀𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87ralbii 3009 . . . 4 (∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
96, 8bitri 264 . . 3 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
105, 9sylib 208 . 2 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
11 dftr2 4787 . . 3 (Tr 𝑧 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1211ralbii 3009 . 2 (∀𝑧𝐴 Tr 𝑧 ↔ ∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
13 df-po 5064 . . 3 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
14 epel 5061 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
15 epel 5061 . . . . . . . 8 (𝑦 E 𝑧𝑦𝑧)
1614, 15anbi12i 733 . . . . . . 7 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
17 epel 5061 . . . . . . 7 (𝑥 E 𝑧𝑥𝑧)
1816, 17imbi12i 339 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
19 elirrv 8542 . . . . . . . 8 ¬ 𝑥𝑥
20 epel 5061 . . . . . . . 8 (𝑥 E 𝑥𝑥𝑥)
2119, 20mtbir 312 . . . . . . 7 ¬ 𝑥 E 𝑥
2221biantrur 526 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2318, 22bitr3i 266 . . . . 5 (((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2423ralbii 3009 . . . 4 (∀𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
25242ralbii 3010 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2613, 25bitr4i 267 . 2 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
2710, 12, 263imtr4i 281 1 (∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wral 2941   class class class wbr 4685  Tr wtr 4785   E cep 5057   Po wpo 5062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064
This theorem is referenced by: (None)
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