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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 2049 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | iota1 5210 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
3 | eqcom 2191 | . . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
4 | 2, 3 | bitrdi 196 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
5 | abid 2177 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | vex 2755 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | elsn 3623 | . . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) |
8 | 4, 5, 7 | 3bitr4g 223 | . . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
9 | 1, 8 | alrimi 1533 | . 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
10 | nfab1 2334 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
11 | nfiota1 5198 | . . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
12 | 11 | nfsn 3667 | . . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
13 | 10, 12 | cleqf 2357 | . 2 ⊢ ({𝑥 ∣ 𝜑} = {(℩𝑥𝜑)} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
14 | 9, 13 | sylibr 134 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1362 = wceq 1364 ∃!weu 2038 ∈ wcel 2160 {cab 2175 {csn 3607 ℩cio 5194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-sn 3613 df-pr 3614 df-uni 3825 df-iota 5196 |
This theorem is referenced by: snriota 5881 |
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