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| Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem3 | Structured version Visualization version GIF version | ||
| Description: Lemma for currysetALT 36951. (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 36949 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
| 3 | 1 | currysetlem1 36948 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
| 4 | 2, 3 | mpbird 257 | . . 3 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑋) |
| 5 | 1 | currysetlem2 36949 | . . . 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 1927 | . . . 4 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝑥 → 𝜑)) |
| 9 | bj-abv 36907 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = V) | |
| 10 | 1, 9 | eqtrid 2789 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝑥 → 𝜑) → 𝑋 = V) |
| 11 | 8, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑋 = V) |
| 12 | nvel 5316 | . . . 4 ⊢ ¬ V ∈ 𝑉 | |
| 13 | eleq1 2829 | . . . 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 |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 |
| This theorem is referenced by: currysetALT 36951 |
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