This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem exnelv

Description: For any set x , there is a set not contained in x . The proof is based on Russell's paradox. (Contributed by NM, 23-Aug-1993) Remove use of ax-12 and ax-13 . (Revised by BJ, 31-May-2019) Extract from nalset . (Revised by Matthew House, 12-Apr-2026)

Ref Expression
Assertion exnelv ∃ 𝑦 ¬ 𝑦 ∈ 𝑥

Proof

Step Hyp Ref Expression
1 ax-sep ⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) )
2 elequ1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) )
3 elequ2 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦 ) )
4 3 notbid ⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦 ) )
5 2 4 anbi12d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) )
6 5 bibi2d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) ↔ ( 𝑧 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) ) )
7 pclem6 ⊢ ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦 ) ) → ¬ 𝑦 ∈ 𝑥 )
8 6 7 biimtrdi ⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ¬ 𝑦 ∈ 𝑥 ) )
9 8 spimvw ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ¬ 𝑦 ∈ 𝑥 )
10 1 9 eximii ⊢ ∃ 𝑦 ¬ 𝑦 ∈ 𝑥