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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 2011 | . . 3 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | iota1 5110 | . . . . 5 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
3 | eqcom 2142 | . . . . 5 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
4 | 2, 3 | syl6bb 195 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
5 | abid 2128 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
6 | vex 2692 | . . . . 5 ⊢ 𝑥 ∈ V | |
7 | 6 | elsn 3548 | . . . 4 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) |
8 | 4, 5, 7 | 3bitr4g 222 | . . 3 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
9 | 1, 8 | alrimi 1503 | . 2 ⊢ (∃!𝑥𝜑 → ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
10 | nfab1 2284 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
11 | nfiota1 5098 | . . . 4 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
12 | 11 | nfsn 3591 | . . 3 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
13 | 10, 12 | cleqf 2306 | . 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 1330 = wceq 1332 ∈ wcel 1481 ∃!weu 2000 {cab 2126 {csn 3532 ℩cio 5094 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-sn 3538 df-pr 3539 df-uni 3745 df-iota 5096 |
This theorem is referenced by: snriota 5767 |
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