MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rele Structured version   Visualization version   GIF version

Theorem rele 5818
Description: The membership relation is a relation. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
rele Rel E

Proof of Theorem rele
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-eprel 5571 . 2 E = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑦}
21relopabiv 5811 1 Rel E
Colors of variables: wff setvar class
Syntax hints:   E cep 5570  Rel wrel 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958  df-opab 5202  df-eprel 5571  df-xp 5673  df-rel 5674
This theorem is referenced by:  bj-epelg  36440  bj-epelb  36441  cnambfre  37030  brcnvep  37627
  Copyright terms: Public domain W3C validator