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Theorem currysetlem3 34385
 Description: Lemma for currysetALT 34386. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)
Hypothesis
Ref Expression
currysetlem2.def 𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}
Assertion
Ref Expression
currysetlem3 ¬ 𝑋𝑉
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem currysetlem3
StepHypRef Expression
1 currysetlem2.def . . . . 5 𝑋 = {𝑥 ∣ (𝑥𝑥𝜑)}
21currysetlem2 34384 . . . 4 (𝑋𝑉 → (𝑋𝑋𝜑))
31currysetlem1 34383 . . . 4 (𝑋𝑉 → (𝑋𝑋 ↔ (𝑋𝑋𝜑)))
42, 3mpbird 260 . . 3 (𝑋𝑉𝑋𝑋)
51currysetlem2 34384 . . . 4 (𝑋𝑋 → (𝑋𝑋𝜑))
65pm2.43i 52 . . 3 (𝑋𝑋𝜑)
7 ax-1 6 . . . . 5 (𝜑 → (𝑥𝑥𝜑))
87alrimiv 1928 . . . 4 (𝜑 → ∀𝑥(𝑥𝑥𝜑))
9 bj-abv 34348 . . . . 5 (∀𝑥(𝑥𝑥𝜑) → {𝑥 ∣ (𝑥𝑥𝜑)} = V)
101, 9syl5eq 2848 . . . 4 (∀𝑥(𝑥𝑥𝜑) → 𝑋 = V)
118, 10syl 17 . . 3 (𝜑𝑋 = V)
12 nvel 5187 . . . 4 ¬ V ∈ 𝑉
13 eleq1 2880 . . . 4 (𝑋 = V → (𝑋𝑉 ↔ V ∈ 𝑉))
1412, 13mtbiri 330 . . 3 (𝑋 = V → ¬ 𝑋𝑉)
154, 6, 11, 144syl 19 . 2 (𝑋𝑉 → ¬ 𝑋𝑉)
1615bj-pm2.01i 34012 1 ¬ 𝑋𝑉
 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:  currysetALT  34386
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