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Mirrors > Home > MPE Home > Th. List > sniota | Structured version Visualization version 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 2582 | . 2 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfab1 2905 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfiota1 6497 | . . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
4 | 3 | nfsn 4711 | . 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
5 | iota1 6520 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
6 | eqcom 2739 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
7 | 5, 6 | bitrdi 286 | . . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
8 | abid 2713 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | velsn 4644 | . . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) | |
10 | 7, 8, 9 | 3bitr4g 313 | . 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
11 | 1, 2, 4, 10 | eqrd 4001 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∃!weu 2562 {cab 2709 {csn 4628 ℩cio 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-v 3476 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 |
This theorem is referenced by: snriota 7401 |
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