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Mirrors > Home > MPE Home > Th. List > Mathboxes > currysetlem1 | Structured version Visualization version GIF version |
Description: Lemma for currysetALT 34283. (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 2829 | . . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} = 𝑋 |
3 | 2 | eleq2i 2903 | . 2 ⊢ (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ 𝑋 ∈ 𝑋) |
4 | nfab1 2978 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} | |
5 | 1, 4 | nfcxfr 2974 | . . 3 ⊢ Ⅎ𝑥𝑋 |
6 | 5, 5 | nfel 2991 | . . . 4 ⊢ Ⅎ𝑥 𝑋 ∈ 𝑋 |
7 | nfv 1914 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
8 | 6, 7 | nfim 1896 | . . 3 ⊢ Ⅎ𝑥(𝑋 ∈ 𝑋 → 𝜑) |
9 | id 22 | . . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
10 | 9, 9 | eleq12d 2906 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝑥 ↔ 𝑋 ∈ 𝑋)) |
11 | 10 | imbi1d 344 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝑥 → 𝜑) ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
12 | 5, 8, 11 | elabgf 3660 | . 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ {𝑥 ∣ (𝑥 ∈ 𝑥 → 𝜑)} ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
13 | 3, 12 | syl5bbr 287 | 1 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ 𝑋 ↔ (𝑋 ∈ 𝑋 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 {cab 2798 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 |
This theorem is referenced by: currysetlem2 34281 currysetlem3 34282 |
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