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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 2030 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | iota1 5172 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
3 | eqcom 2172 | . . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
4 | 2, 3 | bitrdi 195 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
5 | abid 2158 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | vex 2733 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | elsn 3597 | . . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) |
8 | 4, 5, 7 | 3bitr4g 222 | . . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
9 | 1, 8 | alrimi 1515 | . 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
10 | nfab1 2314 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
11 | nfiota1 5160 | . . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
12 | 11 | nfsn 3641 | . . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
13 | 10, 12 | cleqf 2337 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
14 | 9, 13 | sylibr 133 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 = wceq 1348 ∃!weu 2019 ∈ wcel 2141 {cab 2156 {csn 3581 ℩cio 5156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3587 df-pr 3588 df-uni 3795 df-iota 5158 |
This theorem is referenced by: snriota 5835 |
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