Theorem eqid1 27877

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.

Step	Hyp	Ref	Expression
1		biid 253	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
2	1	eqriv 2822	1 ⊢ 𝐴 = 𝐴

Syntax hints:   = wceq 1656   ∈ wcel 2164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-cleq 2818

This theorem is referenced by: (None)