Theorem curryset 37610

Description: Curry's paradox in set theory. This can be seen as a generalization of Russell's paradox, which corresponds to the case where 𝜑 is ⊥. See alternate exposal of basically the same proof currysetALT 37614. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)

Assertion

Ref	Expression
curryset	⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem curryset

Step	Hyp	Ref	Expression
1		currysetlem 37609	. . . . . 6 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑)))
2	1	ibi 270	. . . . 5 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑))
3	2	pm2.43i 53	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑)
4		currysetlem 37609	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑)))
5	3, 4	mpbiri 261	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)})
6		ax-1 6	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑥 → 𝜑))
7	6	alrimiv 1956	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑))
8		bj-abv 37569	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
9	7, 8	syl 18	. . 3 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
10		nvel 5281	. . . 4 ⊢ ¬ V ∈ 𝑉
11		eleq1 2850	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 ↔ V ∈ 𝑉))
12	10, 11	mtbiri 330	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V → ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉)
13	5, 3, 9, 12	4syl 20	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 → ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉)
14	13	pm2.01i 191	1 ⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567   = wceq 1569   ∈ wcel 2142  {cab 2740  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-v 3456

This theorem is used by: (None)