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Metamath Proof Explorer

Theorem nalset

Description: No set contains all sets. Theorem 41 of Suppes p. 30. (Contributed by NM, 23-Aug-1993) Extract exnelv . (Revised by Matthew House, 12-Apr-2026)

		Ref	Expression
	Assertion	nalset	⊢ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥

Proof

Step	Hyp	Ref	Expression
1		alexn	⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥 )
2		exnelv	⊢ ∃ 𝑦 ¬ 𝑦 ∈ 𝑥
3	1 2	mpgbi	⊢ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥