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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 2468 | . 2 ⊢ Ⅎ𝑥∃!𝑥𝜑 | |
2 | nfab1 2753 | . 2 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
3 | nfiota1 5770 | . . 3 ⊢ Ⅎ𝑥(℩𝑥𝜑) | |
4 | 3 | nfsn 4189 | . 2 ⊢ Ⅎ𝑥{(℩𝑥𝜑)} |
5 | iota1 5782 | . . . 4 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ (℩𝑥𝜑) = 𝑥)) | |
6 | eqcom 2617 | . . . 4 ⊢ ((℩𝑥𝜑) = 𝑥 ↔ 𝑥 = (℩𝑥𝜑)) | |
7 | 5, 6 | syl6bb 275 | . . 3 ⊢ (∃!𝑥𝜑 → (𝜑 ↔ 𝑥 = (℩𝑥𝜑))) |
8 | abid 2598 | . . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) | |
9 | velsn 4141 | . . 3 ⊢ (𝑥 ∈ {(℩𝑥𝜑)} ↔ 𝑥 = (℩𝑥𝜑)) | |
10 | 7, 8, 9 | 3bitr4g 302 | . 2 ⊢ (∃!𝑥𝜑 → (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {(℩𝑥𝜑)})) |
11 | 1, 2, 4, 10 | eqrd 3586 | 1 ⊢ (∃!𝑥𝜑 → {𝑥 ∣ 𝜑} = {(℩𝑥𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 {cab 2596 {csn 4125 ℩cio 5766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-sbc 3403 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 |
This theorem is referenced by: snriota 6540 |
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