Theorem curryset 35135

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 35139. (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 35134	. . . . . 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 266	. . . . 5 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑))
3	2	pm2.43i 52	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑)
4		currysetlem 35134	. . . 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 257	. . 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 1930	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑))
8		bj-abv 35091	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
10		nvel 5240	. . . 4 ⊢ ¬ V ∈ 𝑉
11		eleq1 2826	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V → ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 ↔ V ∈ 𝑉))
12	10, 11	mtbiri 327	. . 3 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V → ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉)
13	5, 3, 9, 12	4syl 19	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 → ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉)
14	13	bj-pm2.01i 34743	1 ⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537   = wceq 1539   ∈ wcel 2106  {cab 2715  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-v 3434

This theorem is referenced by: (None)