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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 2673 | . 2 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfab1 2982 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfiota1 6319 | . . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
4 | 3 | nfsn 4646 | . 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
5 | iota1 6335 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
6 | eqcom 2831 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
7 | 5, 6 | syl6bb 289 | . . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
8 | abid 2806 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | velsn 4586 | . . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) | |
10 | 7, 8, 9 | 3bitr4g 316 | . 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
11 | 1, 2, 4, 10 | eqrd 3989 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∃!weu 2652 {cab 2802 {csn 4570 ℩cio 6315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-sbc 3776 df-un 3944 df-in 3946 df-ss 3955 df-sn 4571 df-pr 4573 df-uni 4842 df-iota 6317 |
This theorem is referenced by: snriota 7150 fnimasnd 39127 |
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