Theorem nalset 5267

Description: No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5266. (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 1877	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥)
2		exnelv 5266	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
3	1, 2	mpgbi 1830	1 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-sep 5248

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812

This theorem is used by:  vnexOLD  5271  iota0ndef  48152  aiota0ndef  48210