Theorem elirrvALT

Description: Alternate proof of elirrv , shorter but using more axioms. (Contributed by BTernaryTau, 28-Dec-2025) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	elirrvALT	⊢ ¬ 𝑥 ∈ 𝑥

Proof

Step	Hyp	Ref	Expression
1		zfregfr	⊢ E Fr 𝑥
2		efrirr	⊢ ( E Fr 𝑥 → ¬ 𝑥 ∈ 𝑥 )
3	1 2	ax-mp	⊢ ¬ 𝑥 ∈ 𝑥