Mathbox for Jeff Hankins < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnerel Structured version   Visualization version   GIF version

Theorem fnerel 33760
 Description: Fineness is a relation. (Contributed by Jeff Hankins, 28-Sep-2009.)
Assertion
Ref Expression
fnerel Rel Fne

Proof of Theorem fnerel
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fne 33759 . 2 Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}
21relopabi 5671 1 Rel Fne
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∀wral 3130   ∩ cin 3907   ⊆ wss 3908  𝒫 cpw 4511  ∪ cuni 4813  Rel wrel 5537  Fnecfne 33758 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-opab 5105  df-xp 5538  df-rel 5539  df-fne 33759 This theorem is referenced by:  isfne  33761  isfne4  33762  fnetr  33773  fneval  33774  fneer  33775  fnessref  33779
 Copyright terms: Public domain W3C validator