Theorem exnelv 5281

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 2216 and ax-13 2407. (Revised by BJ, 31-May-2019.) Extract from nalset 5282. (Revised by Matthew House, 12-Apr-2026.)

Assertion

Ref	Expression
exnelv	⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem exnelv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-sep 5262	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))
2		elequ1 2153	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3		elequ2 2161	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦))
4	3	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
5	2, 4	anbi12d 644	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)))
6	5	bibi2d 345	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) ↔ (𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))))
7		pclem6 1043	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑦 ∈ 𝑥)
8	6, 7	biimtrdi 256	. . 3 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥))
9	8	spimvw 2019	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥)
10	1, 9	eximii 1870	1 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401  ∀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 5262

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

This theorem is used by:  nalset  5282  vneqv  5284  kmlem2  10154  mh-infprim1bi  37098