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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 2587 | . 2 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
| 2 | nfab1 2906 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
| 3 | nfiota1 6515 | . . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
| 4 | 3 | nfsn 4706 | . 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} | 
| 5 | iota1 6537 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
| 6 | eqcom 2743 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
| 7 | 5, 6 | bitrdi 287 | . . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) | 
| 8 | abid 2717 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
| 9 | velsn 4641 | . . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) | |
| 10 | 7, 8, 9 | 3bitr4g 314 | . 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) | 
| 11 | 1, 2, 4, 10 | eqrd 4002 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∃!weu 2567 {cab 2713 {csn 4625 ℩cio 6511 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-v 3481 df-un 3955 df-ss 3967 df-sn 4626 df-pr 4628 df-uni 4907 df-iota 6513 | 
| This theorem is referenced by: snriota 7422 | 
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