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Theorem untangtr 31717
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 4786 . . . 4 (Tr 𝐴 𝐴𝐴)
2 ssralv 3699 . . . 4 ( 𝐴𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
31, 2sylbi 207 . . 3 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
4 elequ1 2037 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
5 elequ2 2044 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
64, 5bitrd 268 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 307 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
87cbvralv 3201 . . . 4 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦 𝐴 ¬ 𝑦𝑦)
9 untuni 31712 . . . 4 (∀𝑦 𝐴 ¬ 𝑦𝑦 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
108, 9bitri 264 . . 3 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
113, 10syl6ib 241 . 2 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
12 untelirr 31711 . . 3 (∀𝑦𝑥 ¬ 𝑦𝑦 → ¬ 𝑥𝑥)
1312ralimi 2981 . 2 (∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑥𝐴 ¬ 𝑥𝑥)
1411, 13impbid1 215 1 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wral 2941  wss 3607   cuni 4468  Tr wtr 4785
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-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-uni 4469  df-tr 4786
This theorem is referenced by: (None)
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