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Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem3 | Structured version Visualization version GIF version |
Description: Lemma for currysetALT 34833. (Contributed by BJ, 23-Sep-2023.) This proof is intuitionistically valid. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
currysetlem2.def | ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} |
Ref | Expression |
---|---|
currysetlem3 | ⊢ ¬ 𝑋 ∈ 𝑉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | currysetlem2.def | . . . . 5 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
2 | 1 | currysetlem2 34831 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
3 | 1 | currysetlem1 34830 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
4 | 2, 3 | mpbird 260 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑋) |
5 | 1 | currysetlem2 34831 | . . . 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 1935 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑)) |
9 | bj-abv 34785 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V) | |
10 | 1, 9 | syl5eq 2786 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → 𝑋 = V) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 = V) |
12 | nvel 5198 | . . . 4 ⊢ ¬ V ∈ 𝑉 | |
13 | eleq1 2821 | . . . 4 ⊢ (𝑋 = V → (𝑋 ∈ 𝑉 ↔ V ∈ 𝑉)) | |
14 | 12, 13 | mtbiri 330 | . . 3 ⊢ (𝑋 = V → ¬ 𝑋 ∈ 𝑉) |
15 | 4, 6, 11, 14 | 4syl 19 | . 2 ⊢ (𝑋 ∈ 𝑉 → ¬ 𝑋 ∈ 𝑉) |
16 | 15 | bj-pm2.01i 34437 | 1 ⊢ ¬ 𝑋 ∈ 𝑉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1541 = wceq 1543 ∈ wcel 2110 {cab 2712 Vcvv 3401 |
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 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 |
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 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-v 3403 |
This theorem is referenced by: currysetALT 34833 |
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