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Theorem untuni 32943
 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 3266 . . . 4 (∀𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
21albii 1820 . . 3 (∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
3 ralcom4 3222 . . 3 (∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥))
4 eluni2 4818 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
54imbi1i 352 . . . 4 ((𝑥 𝐴 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
65albii 1820 . . 3 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
72, 3, 63bitr4ri 306 . 2 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
8 df-ral 3130 . 2 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥))
9 df-ral 3130 . . 3 (∀𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
109ralbii 3152 . 2 (∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
117, 8, 103bitr4i 305 1 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1535   ∈ wcel 2114  ∀wral 3125  ∃wrex 3126  ∪ cuni 4814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-v 3475  df-uni 4815 This theorem is referenced by:  untangtr  32948  dfon2lem3  33038  dfon2lem7  33042
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