MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqid1 Unicode version

Theorem eqid1 20840
Description: Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
eqid1  |-  A  =  A

Proof of Theorem eqid1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 biid 227 . 2  |-  ( x  e.  A  <->  x  e.  A )
21eqriv 2280 1  |-  A  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-cleq 2276
  Copyright terms: Public domain W3C validator