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Theorem dfatprc 47147
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 3506 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
21orcd 873 . 2 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 ianor 983 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
4 df-dfat 47136 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
53, 4xchnxbir 333 . 2 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
62, 5sylibr 234 1 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  wcel 2107  Vcvv 3479  {csn 4625  dom cdm 5684  cres 5686  Fun wfun 6554   defAt wdfat 47133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dfat 47136
This theorem is referenced by: (None)
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