Theorem ruOLD

Description: Obsolete version of ru as of 20-Jun-2025. (Contributed by NM, 7-Aug-1994) Remove use of ax-13 . (Revised by BJ, 12-Oct-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	ruOLD	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } ∉ V

Proof

Step	Hyp	Ref	Expression
1		pm5.19	⊢ ¬ ( 𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦 )
2		eleq1w	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦 ) )
3		df-nel	⊢ ( 𝑥 ∉ 𝑥 ↔ ¬ 𝑥 ∈ 𝑥 )
4		id	⊢ ( 𝑥 = 𝑦 → 𝑥 = 𝑦 )
5	4 4	eleq12d	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦 ) )
6	5	notbid	⊢ ( 𝑥 = 𝑦 → ( ¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦 ) )
7	3 6	bitrid	⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∉ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦 ) )
8	2 7	bibi12d	⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥 ) ↔ ( 𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦 ) ) )
9	8	spvv	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥 ) → ( 𝑦 ∈ 𝑦 ↔ ¬ 𝑦 ∈ 𝑦 ) )
10	1 9	mto	⊢ ¬ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥 )
11		eqabb	⊢ ( 𝑦 = { 𝑥 ∣ 𝑥 ∉ 𝑥 } ↔ ∀ 𝑥 ( 𝑥 ∈ 𝑦 ↔ 𝑥 ∉ 𝑥 ) )
12	10 11	mtbir	⊢ ¬ 𝑦 = { 𝑥 ∣ 𝑥 ∉ 𝑥 }
13	12	nex	⊢ ¬ ∃ 𝑦 𝑦 = { 𝑥 ∣ 𝑥 ∉ 𝑥 }
14		isset	⊢ ( { 𝑥 ∣ 𝑥 ∉ 𝑥 } ∈ V ↔ ∃ 𝑦 𝑦 = { 𝑥 ∣ 𝑥 ∉ 𝑥 } )
15	13 14	mtbir	⊢ ¬ { 𝑥 ∣ 𝑥 ∉ 𝑥 } ∈ V
16	15	nelir	⊢ { 𝑥 ∣ 𝑥 ∉ 𝑥 } ∉ V