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Theorem dfatprc 41984
Description: A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)
Assertion
Ref Expression
dfatprc 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Proof of Theorem dfatprc
StepHypRef Expression
1 prcnel 3406 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
21orcd 900 . 2 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 ianor 1005 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
4 df-dfat 41973 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
53, 4xchnxbir 325 . 2 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
62, 5sylibr 226 1 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874  wcel 2157  Vcvv 3385  {csn 4368  dom cdm 5312  cres 5314  Fun wfun 6095   defAt wdfat 41970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-v 3387  df-dfat 41973
This theorem is referenced by: (None)
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