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Metamath Proof Explorer

Theorem wfelirr

Description: A well-founded set is not a member of itself. This proof does not require the axiom of regularity, unlike elirr . (Contributed by Mario Carneiro, 2-Jan-2017)

		Ref	Expression
	Assertion	wfelirr	⊢ ( 𝐴 ∈ ∪ ( 𝑅₁ “ On ) → ¬ 𝐴 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rankon	⊢ ( rank ‘ 𝐴 ) ∈ On
2	1	onirri	⊢ ¬ ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐴 )
3		rankelb	⊢ ( 𝐴 ∈ ∪ ( 𝑅₁ “ On ) → ( 𝐴 ∈ 𝐴 → ( rank ‘ 𝐴 ) ∈ ( rank ‘ 𝐴 ) ) )
4	2 3	mtoi	⊢ ( 𝐴 ∈ ∪ ( 𝑅₁ “ On ) → ¬ 𝐴 ∈ 𝐴 )