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Theorem untangtr 30638
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 4675 . . . 4 (Tr 𝐴 𝐴𝐴)
2 ssralv 3628 . . . 4 ( 𝐴𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
31, 2sylbi 205 . . 3 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥 𝐴 ¬ 𝑥𝑥))
4 elequ1 1983 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
5 elequ2 1990 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
64, 5bitrd 266 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
76notbid 306 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
87cbvralv 3146 . . . 4 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦 𝐴 ¬ 𝑦𝑦)
9 untuni 30633 . . . 4 (∀𝑦 𝐴 ¬ 𝑦𝑦 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
108, 9bitri 262 . . 3 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦)
113, 10syl6ib 239 . 2 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
12 untelirr 30632 . . 3 (∀𝑦𝑥 ¬ 𝑦𝑦 → ¬ 𝑥𝑥)
1312ralimi 2935 . 2 (∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦 → ∀𝑥𝐴 ¬ 𝑥𝑥)
1411, 13impbid1 213 1 (Tr 𝐴 → (∀𝑥𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥𝐴𝑦𝑥 ¬ 𝑦𝑦))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wral 2895  wss 3539   cuni 4366  Tr wtr 4674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367  df-tr 4675
This theorem is referenced by: (None)
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