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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 1954 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | iota1 4932 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
3 | eqcom 2085 | . . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
4 | 2, 3 | syl6bb 194 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
5 | abid 2071 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | vex 2614 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | elsn 3433 | . . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) |
8 | 4, 5, 7 | 3bitr4g 221 | . . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
9 | 1, 8 | alrimi 1456 | . 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
10 | nfab1 2225 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
11 | nfiota1 4920 | . . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
12 | 11 | nfsn 3471 | . . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
13 | 10, 12 | cleqf 2246 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
14 | 9, 13 | sylibr 132 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1283 = wceq 1285 ∈ wcel 1434 ∃!weu 1943 {cab 2069 {csn 3417 ℩cio 4916 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-rex 2359 df-v 2613 df-sbc 2826 df-un 2987 df-sn 3423 df-pr 3424 df-uni 3623 df-iota 4918 |
This theorem is referenced by: snriota 5550 |
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