Theorem bj-ru

Description: Remove dependency on ax-13 (and df-v ) from Russell's paradox ru expressed with primitive symbols and with a class variable V . Note the more economical use of bj-elissetv instead of isset to avoid use of df-v . (Contributed by BJ, 12-Oct-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-ru	⊢ ¬ { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝑉

Proof

Step	Hyp	Ref	Expression
1		bj-ru1	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 }
2		bj-elissetv	⊢ ( { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝑉 → ∃ 𝑦 𝑦 = { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } )
3	1 2	mto	⊢ ¬ { 𝑥 ∣ ¬ 𝑥 ∈ 𝑥 } ∈ 𝑉