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Theorem curryset 34382
 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 34386. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
Assertion
Ref Expression
curryset ¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem curryset
StepHypRef Expression
1 currysetlem 34381 . . . . . 6 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → 𝜑)))
21ibi 270 . . . . 5 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → 𝜑))
32pm2.43i 52 . . . 4 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → 𝜑)
4 currysetlem 34381 . . . 4 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉 → ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} ↔ ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)} → 𝜑)))
53, 4mpbiri 261 . . 3 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉 → {𝑥 ∣ (𝑥𝑥𝜑)} ∈ {𝑥 ∣ (𝑥𝑥𝜑)})
6 ax-1 6 . . . . 5 (𝜑 → (𝑥𝑥𝜑))
76alrimiv 1928 . . . 4 (𝜑 → ∀𝑥(𝑥𝑥𝜑))
8 bj-abv 34348 . . . 4 (∀𝑥(𝑥𝑥𝜑) → {𝑥 ∣ (𝑥𝑥𝜑)} = V)
97, 8syl 17 . . 3 (𝜑 → {𝑥 ∣ (𝑥𝑥𝜑)} = V)
10 nvel 5187 . . . 4 ¬ V ∈ 𝑉
11 eleq1 2880 . . . 4 ({𝑥 ∣ (𝑥𝑥𝜑)} = V → ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉 ↔ V ∈ 𝑉))
1210, 11mtbiri 330 . . 3 ({𝑥 ∣ (𝑥𝑥𝜑)} = V → ¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉)
135, 3, 9, 124syl 19 . 2 ({𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉 → ¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉)
1413bj-pm2.01i 34012 1 ¬ {𝑥 ∣ (𝑥𝑥𝜑)} ∈ 𝑉
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536   = wceq 1538   ∈ wcel 2112  {cab 2779  Vcvv 3444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446 This theorem is referenced by: (None)
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