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Metamath Proof Explorer

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