Theorem nalset 5277

Description: No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Extract exnelv 5276. (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 1875	. 2 ⊢ (∀𝑥∃𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥)
2		exnelv 5276	. 2 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥
3	1, 2	mpgbi 1828	1 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3  ∀wal 1568  ∃wex 1809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  vnexOLD  5281  iota0ndef  47776  aiota0ndef  47834