Theorem exnelv 5262

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 2211 and ax-13 2402. (Revised by BJ, 31-May-2019.) Extract from nalset 5263. (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 5245	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))
2		elequ1 2148	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3		elequ2 2156	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦))
4	3	notbid 320	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
5	2, 4	anbi12d 641	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)))
6	5	bibi2d 344	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) ↔ (𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))))
7		pclem6 1038	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑦 ∈ 𝑥)
8	6, 7	biimtrdi 255	. . 3 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥))
9	8	spimvw 2005	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥)
10	1, 9	eximii 1856	1 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 399  ∀wal 1557  ∃wex 1798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-sep 5245

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799

This theorem is referenced by:  nalset  5263  vneqv  5265  kmlem2  10105  mh-infprim1bi  36870