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Theorem untuni 31712
Description: The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
untuni (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem untuni
StepHypRef Expression
1 r19.23v 3052 . . . 4 (∀𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
21albii 1787 . . 3 (∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
3 ralcom4 3255 . . 3 (∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥))
4 eluni2 4472 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
54imbi1i 338 . . . 4 ((𝑥 𝐴 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
65albii 1787 . . 3 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
72, 3, 63bitr4ri 293 . 2 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
8 df-ral 2946 . 2 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥))
9 df-ral 2946 . . 3 (∀𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
109ralbii 3009 . 2 (∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
117, 8, 103bitr4i 292 1 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1521  wcel 2030  wral 2941  wrex 2942   cuni 4468
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-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-uni 4469
This theorem is referenced by:  untangtr  31717  dfon2lem3  31814  dfon2lem7  31818
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