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Theorem untangtr 33050
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 5137 . . . 4 (Tr 𝐴 𝐴𝐴)
2 ssralv 3981 . . . 4 ( 𝐴𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
31, 2sylbi 220 . . 3 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
4 elequ1 2118 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
5 elequ2 2126 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
64, 5bitrd 282 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 321 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
87cbvralvw 3396 . . . 4 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦 𝐴 ¬ 𝑦𝑦)
9 untuni 33045 . . . 4 (∀𝑦 𝐴 ¬ 𝑦𝑦 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
108, 9bitri 278 . . 3 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
113, 10syl6ib 254 . 2 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
12 untelirr 33044 . . 3 (∀𝑦𝑥 ¬ 𝑦𝑦 → ¬ 𝑥𝑥)
1312ralimi 3128 . 2 (∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑥𝐴 ¬ 𝑥𝑥)
1411, 13impbid1 228 1 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wral 3106  wss 3881   cuni 4800  Tr wtr 5136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-tr 5137
This theorem is referenced by: (None)
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