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Theorem untuni 34678
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 3183 . . . 4 (∀𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
21albii 1822 . . 3 (∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
3 ralcom4 3284 . . 3 (∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥) ↔ ∀𝑥𝑦𝐴 (𝑥𝑦 → ¬ 𝑥𝑥))
4 eluni2 4913 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
54imbi1i 350 . . . 4 ((𝑥 𝐴 → ¬ 𝑥𝑥) ↔ (∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
65albii 1822 . . 3 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑥(∃𝑦𝐴 𝑥𝑦 → ¬ 𝑥𝑥))
72, 3, 63bitr4ri 304 . 2 (∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥) ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
8 df-ral 3063 . 2 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥 𝐴 → ¬ 𝑥𝑥))
9 df-ral 3063 . . 3 (∀𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
109ralbii 3094 . 2 (∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥(𝑥𝑦 → ¬ 𝑥𝑥))
117, 8, 103bitr4i 303 1 (∀𝑥 𝐴 ¬ 𝑥𝑥 ↔ ∀𝑦𝐴𝑥𝑦 ¬ 𝑥𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1540  wcel 2107  wral 3062  wrex 3071   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-uni 4910
This theorem is referenced by:  untangtr  34683  dfon2lem3  34757  dfon2lem7  34761
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