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| Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem2 | 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 |
|---|---|
| currysetlem2 | ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | currysetlem2.def | . . . 4 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
| 2 | 1 | currysetlem1 36948 | . . 3 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
| 3 | 2 | biimpd 229 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → (𝑋 ∈ 𝑋 → 𝜑))) |
| 4 | 3 | pm2.43d 53 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 → 𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cab 2714 |
| 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 |
| 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: currysetlem3 36950 |
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