Theorem bj-ru1

Description: A version of Russell's paradox ru (see also bj-ru ). (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ru1	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		bj-ru0	⊢ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥 )
2		abeq2	⊢ ( 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ ¬ 𝑥 ∈ 𝑥 ) )
3	1 2	mtbir	⊢ ¬ 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }
4	3	nex	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }