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Mirrors > Home > MPE Home > Th. List > eusn | Structured version Visualization version GIF version |
Description: Two ways to express "𝐴 is a singleton." (Contributed by NM, 30-Oct-2010.) |
Ref | Expression |
---|---|
eusn | ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn 4655 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥}) | |
2 | abid2 2957 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | eqeq1i 2826 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥}) |
4 | 3 | exbii 1844 | . 2 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}) |
5 | 1, 4 | bitri 277 | 1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃!weu 2649 {cab 2799 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-sn 4561 |
This theorem is referenced by: initoid 17259 termoid 17260 initoeu2lem1 17268 funpartfv 33401 irinitoringc 44334 |
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