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Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem1 | Structured version Visualization version GIF version |
Description: Lemma for currysetALT 34876. (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 |
---|---|
currysetlem1 | ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | currysetlem2.def | . . . 4 ⊢ 𝑋 = {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
2 | 1 | eqcomi 2746 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = 𝑋 |
3 | 2 | eleq2i 2829 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ 𝑋 ∈ 𝑋) |
4 | nfab1 2906 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
5 | 1, 4 | nfcxfr 2902 | . . 3 ⊢ Ⅎ𝑥𝑋 |
6 | 5, 5 | nfel 2918 | . . . 4 ⊢ Ⅎ𝑥 𝑋 ∈ 𝑋 |
7 | nfv 1922 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
8 | 6, 7 | nfim 1904 | . . 3 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑋 → 𝜑) |
9 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
10 | 9, 9 | eleq12d 2832 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑥 ↔ 𝑋 ∈ 𝑋)) |
11 | 10 | imbi1d 345 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑥 → 𝜑) ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
12 | 5, 8, 11 | elabgf 3583 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
13 | 3, 12 | bitr3id 288 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 {cab 2714 |
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 2708 |
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 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-v 3410 |
This theorem is referenced by: currysetlem2 34874 currysetlem3 34875 |
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