Theorem curryset 34821

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 34825. (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 34820	. . . . . 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 52	. . . 4 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} → 𝜑)
4		currysetlem 34820	. . . 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 1935	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑))
8		bj-abv 34778	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
10		nvel 5194	. . . 4 ⊢ ¬ V ∈ 𝑉
11		eleq1 2818	. . . 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 19	. 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉 → ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉)
14	13	bj-pm2.01i 34429	1 ⊢ ¬ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ∈ 𝑉

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1541   = wceq 1543   ∈ wcel 2112  {cab 2714  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-v 3400

This theorem is referenced by: (None)