Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  untangtr Structured version   Visualization version   GIF version

Theorem untangtr 35714
Description: A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
Assertion
Ref Expression
untangtr (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untangtr
StepHypRef Expression
1 df-tr 5260 . . . 4 (Tr 𝐴 𝐴𝐴)
2 ssralv 4052 . . . 4 ( 𝐴𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
31, 2sylbi 217 . . 3 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
4 elequ1 2115 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
5 elequ2 2123 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
64, 5bitrd 279 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 318 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
87cbvralvw 3237 . . . 4 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦 𝐴 ¬ 𝑦𝑦)
9 untuni 35709 . . . 4 (∀𝑦 𝐴 ¬ 𝑦𝑦 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
108, 9bitri 275 . . 3 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
113, 10imbitrdi 251 . 2 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
12 untelirr 35708 . . 3 (∀𝑦𝑥 ¬ 𝑦𝑦 → ¬ 𝑥𝑥)
1312ralimi 3083 . 2 (∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑥𝐴 ¬ 𝑥𝑥)
1411, 13impbid1 225 1 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wral 3061  wss 3951   cuni 4907  Tr wtr 5259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator