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Theorem fnerel 33147
 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 33146 . 2 Fne = {⟨𝑥, 𝑦⟩ ∣ ( 𝑥 = 𝑦 ∧ ∀𝑧𝑥 𝑧 (𝑦 ∩ 𝒫 𝑧))}
21relopabi 5537 1 Rel Fne
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 387   = wceq 1507  ∀wral 3082   ∩ cin 3824   ⊆ wss 3825  𝒫 cpw 4416  ∪ cuni 4706  Rel wrel 5405  Fnecfne 33145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-opab 4986  df-xp 5406  df-rel 5407  df-fne 33146 This theorem is referenced by:  isfne  33148  isfne4  33149  fnetr  33160  fneval  33161  fneer  33162  fnessref  33166
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