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Theorem elpotr 31440
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 2924 . . . . . 6 (∀𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
21alimi 1736 . . . . 5 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
3 alral 2924 . . . . 5 (∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
42, 3syl 17 . . . 4 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
54ralimi 2948 . . 3 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
6 ralcom 3092 . . . 4 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
7 ralcom 3092 . . . . 5 (∀𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87ralbii 2976 . . . 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 4724 . . 3 (Tr 𝑧 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1211ralbii 2976 . 2 (∀𝑧𝐴 Tr 𝑧 ↔ ∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
13 df-po 5005 . . 3 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
14 epel 4998 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
15 epel 4998 . . . . . . . 8 (𝑦 E 𝑧𝑦𝑧)
1614, 15anbi12i 732 . . . . . . 7 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
17 epel 4998 . . . . . . 7 (𝑥 E 𝑧𝑥𝑧)
1816, 17imbi12i 340 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
19 elirrv 8464 . . . . . . . 8 ¬ 𝑥𝑥
20 epel 4998 . . . . . . . 8 (𝑥 E 𝑥𝑥𝑥)
2119, 20mtbir 313 . . . . . . 7 ¬ 𝑥 E 𝑥
2221biantrur 527 . . . . . 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 2976 . . . 4 (∀𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
25242ralbii 2977 . . 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 384  wal 1478  wral 2908   class class class wbr 4623  Tr wtr 4722   E cep 4993   Po wpo 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-reg 8457
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-tr 4723  df-eprel 4995  df-po 5005
This theorem is referenced by: (None)
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