Theorem exnelv 5242

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 2189 and ax-13 2380. (Revised by BJ, 31-May-2019.) Extract from nalset 5243. (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 5225	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧))
2		elequ1 2126	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
3		elequ2 2134	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ 𝑧 ↔ 𝑧 ∈ 𝑦))
4	3	notbid 319	. . . . . 6 ⊢ (𝑧 = 𝑦 → (¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
5	2, 4	anbi12d 638	. . . . 5 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)))
6	5	bibi2d 343	. . . 4 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) ↔ (𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦))))
7		pclem6 1033	. . . 4 ⊢ ((𝑧 ∈ 𝑦 ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑦 ∈ 𝑥)
8	6, 7	biimtrdi 254	. . 3 ⊢ (𝑧 = 𝑦 → ((𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥))
9	8	spimvw 1993	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ (𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧)) → ¬ 𝑦 ∈ 𝑥)
10	1, 9	eximii 1844	1 ⊢ ∃𝑦 ¬ 𝑦 ∈ 𝑥

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-sep 5225

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  nalset  5243  vneqv  5245  kmlem2  10072  mh-infprim1bi  36781