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Mirrors > Home > MPE Home > Th. List > reusn | Structured version Visualization version GIF version |
Description: A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.) |
Ref | Expression |
---|---|
reusn | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 4621 | . 2 ⊢ (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦}) | |
2 | df-reu 3113 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | df-rab 3115 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
4 | 3 | eqeq1i 2803 | . . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦}) |
5 | 4 | exbii 1849 | . 2 ⊢ (∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦} ↔ ∃𝑦{𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = {𝑦}) |
6 | 1, 2, 5 | 3bitr4i 306 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦{𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃!weu 2628 {cab 2776 ∃!wreu 3108 {crab 3110 {csn 4525 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-reu 3113 df-rab 3115 df-sn 4526 |
This theorem is referenced by: reuen1 8561 cshwrepswhash1 16428 frcond3 28054 vdgn1frgrv2 28081 ddemeas 31605 |
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