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Theorem elpotr 32929
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 3159 . . . . . 6 (∀𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
21alimi 1805 . . . . 5 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
3 alral 3159 . . . . 5 (∀𝑥𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
42, 3syl 17 . . . 4 (∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
54ralimi 3165 . . 3 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
6 ralcom 3359 . . . 4 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
7 ralcom 3359 . . . . 5 (∀𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87ralbii 3170 . . . 4 (∀𝑥𝐴𝑧𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
96, 8bitri 276 . . 3 (∀𝑧𝐴𝑥𝐴𝑦𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
105, 9sylib 219 . 2 (∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧) → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
11 dftr2 5171 . . 3 (Tr 𝑧 ↔ ∀𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
1211ralbii 3170 . 2 (∀𝑧𝐴 Tr 𝑧 ↔ ∀𝑧𝐴𝑥𝑦((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
13 df-po 5473 . . 3 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
14 epel 5468 . . . . . . . 8 (𝑥 E 𝑦𝑥𝑦)
15 epel 5468 . . . . . . . 8 (𝑦 E 𝑧𝑦𝑧)
1614, 15anbi12i 626 . . . . . . 7 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
17 epel 5468 . . . . . . 7 (𝑥 E 𝑧𝑥𝑧)
1816, 17imbi12i 352 . . . . . 6 (((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧) ↔ ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
19 elirrv 9054 . . . . . . . 8 ¬ 𝑥𝑥
20 epel 5468 . . . . . . . 8 (𝑥 E 𝑥𝑥𝑥)
2119, 20mtbir 324 . . . . . . 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 278 . . . . 5 (((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2423ralbii 3170 . . . 4 (∀𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
25242ralbii 3171 . . 3 (∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧) ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥 E 𝑥 ∧ ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
2613, 25bitr4i 279 . 2 ( E Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
2710, 12, 263imtr4i 293 1 (∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1528  wral 3143   class class class wbr 5063  Tr wtr 5169   E cep 5463   Po wpo 5471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-reg 9050
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473
This theorem is referenced by: (None)
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