Theorem nalset 5279

Description: No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5278. (Revised by Matthew House, 12-Apr-2026.)

Assertion

Ref	Expression
nalset	⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem nalset

Step	Hyp	Ref	Expression
1		alexn 1878	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥)
2		exnelv 5278	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
3	1, 2	mpgbi 1831	1 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by:  vnexOLD  5283  iota0ndef  47834  aiota0ndef  47892