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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 2667 | . 2 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfab1 2976 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfiota1 6309 | . . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
4 | 3 | nfsn 4635 | . 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
5 | iota1 6325 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
6 | eqcom 2825 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
7 | 5, 6 | syl6bb 288 | . . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
8 | abid 2800 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | velsn 4573 | . . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) | |
10 | 7, 8, 9 | 3bitr4g 315 | . 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
11 | 1, 2, 4, 10 | eqrd 3983 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∃!weu 2646 {cab 2796 {csn 4557 ℩cio 6305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-sbc 3770 df-un 3938 df-sn 4558 df-pr 4560 df-uni 4831 df-iota 6307 |
This theorem is referenced by: snriota 7136 fnimasnd 38999 |
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