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Mirrors > Home > ILE Home > Th. List > sniota | GIF version |
Description: A class abstraction with a unique member can be expressed as a singleton. (Contributed by Mario Carneiro, 23-Dec-2016.) |
Ref | Expression |
---|---|
sniota | ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeu1 1911 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | iota1 4881 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
3 | eqcom 2042 | . . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
4 | 2, 3 | syl6bb 185 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
5 | abid 2028 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | elsn 3391 | . . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) |
8 | 4, 5, 7 | 3bitr4g 212 | . . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
9 | 1, 8 | alrimi 1415 | . 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
10 | nfab1 2180 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
11 | nfiota1 4869 | . . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
12 | 11 | nfsn 3430 | . . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
13 | 10, 12 | cleqf 2201 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
14 | 9, 13 | sylibr 137 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∃!weu 1900 {cab 2026 {csn 3375 ℩cio 4865 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-sn 3381 df-pr 3382 df-uni 3581 df-iota 4867 |
This theorem is referenced by: snriota 5497 |
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