Theorem currysetlem3 36915

Description: Lemma for currysetALT 36916. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.)

Hypothesis

Ref	Expression
currysetlem2.def	⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}

Assertion

Ref	Expression
currysetlem3	⊢ ¬ 𝑋 ∈ 𝑉

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem currysetlem3

Step	Hyp	Ref	Expression
1		currysetlem2.def	. . . . 5 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)}
2	1	currysetlem2 36914	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑))
3	1	currysetlem1 36913	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑)))
4	2, 3	mpbird 257	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑋)
5	1	currysetlem2 36914	. . . 4 ⊢ (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑))
6	5	pm2.43i 52	. . 3 ⊢ (𝑋 ∈ 𝑋 → 𝜑)
7		ax-1 6	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑥 → 𝜑))
8	7	alrimiv 1926	. . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑))
9		bj-abv 36872	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V)
10	1, 9	eqtrid 2792	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → 𝑋 = V)
11	8, 10	syl 17	. . 3 ⊢ (𝜑 → 𝑋 = V)
12		nvel 5334	. . . 4 ⊢ ¬ V ∈ 𝑉
13		eleq1 2832	. . . 4 ⊢ (𝑋 = V → (𝑋 ∈ 𝑉 ↔ V ∈ 𝑉))
14	12, 13	mtbiri 327	. . 3 ⊢ (𝑋 = V → ¬ 𝑋 ∈ 𝑉)
15	4, 6, 11, 14	4syl 19	. 2 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑋 ∈ 𝑉)
16	15	pm2.01i 189	1 ⊢ ¬ 𝑋 ∈ 𝑉

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-v 3490

This theorem is referenced by:  currysetALT  36916