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

Theorem reli 5159
Description: The identity relation is a relation. Part of Exercise 4.12(p) of [Mendelson] p. 235. (Contributed by NM, 26-Apr-1998.) (Revised by Mario Carneiro, 21-Dec-2013.)
Assertion
Ref Expression
reli Rel I

Proof of Theorem reli
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfid3 4944 . 2 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
21relopabi 5156 1 Rel I
Colors of variables: wff setvar class
Syntax hints:   I cid 4938  Rel wrel 5033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035
This theorem is referenced by:  ideqg  5183  issetid  5186  iss  5354  intirr  5420  funi  5820  f1ovi  6072  idssen  7863  idsset  30973  bj-elid  32058
  Copyright terms: Public domain W3C validator