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

Theorem eqid1 30192
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 261 . 2 (𝑥𝐴𝑥𝐴)
21eqriv 2721 1 𝐴 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098
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-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-cleq 2716
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator