Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfatprc Structured version   Visualization version   GIF version

Theorem dfatprc 43322
 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 3519 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
21orcd 869 . 2 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 ianor 978 . . 3 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
4 df-dfat 43311 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
53, 4xchnxbir 335 . 2 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
62, 5sylibr 236 1 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∈ wcel 2110  Vcvv 3495  {csn 4561  dom cdm 5550   ↾ cres 5552  Fun wfun 6344   defAt wdfat 43308 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-v 3497  df-dfat 43311 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator