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Mirrors > Home > MPE Home > Th. List > snriota | Structured version Visualization version GIF version |
Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006.) |
Ref | Expression |
---|---|
snriota | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3365 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
2 | sniota 6545 | . . 3 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))}) |
4 | df-rab 3420 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
5 | df-riota 7380 | . . 3 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 5 | sneqi 4644 | . 2 ⊢ {(℩𝑥 ∈ 𝐴 𝜑)} = {(℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))} |
7 | 3, 4, 6 | 3eqtr4g 2791 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜑} = {(℩𝑥 ∈ 𝐴 𝜑)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∃!weu 2557 {cab 2703 ∃!wreu 3362 {crab 3419 {csn 4633 ℩cio 6504 ℩crio 7379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-un 3952 df-ss 3964 df-sn 4634 df-pr 4636 df-uni 4914 df-iota 6506 df-riota 7380 |
This theorem is referenced by: divalgmod 16408 |
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