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Theorem untuni 30840
 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 3005 . . . 4 (∀𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
21albii 1737 . . 3 (∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
3 ralcom4 3197 . . 3 (∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥))
4 eluni2 4376 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
54imbi1i 338 . . . 4 ((𝑥 𝐴 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
65albii 1737 . . 3 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
72, 3, 63bitr4ri 292 . 2 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
8 df-ral 2901 . 2 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥))
9 df-ral 2901 . . 3 (∀𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
109ralbii 2963 . 2 (∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
117, 8, 103bitr4i 291 1 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-uni 4373 This theorem is referenced by:  untangtr  30845  dfon2lem3  30934  dfon2lem7  30938
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