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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 4403 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥}) | |
2 | abid2 2881 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | eqeq1i 2763 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥}) |
4 | 3 | exbii 1921 | . 2 ⊢ (∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}) |
5 | 1, 4 | bitri 264 | 1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 = wceq 1630 ∃wex 1851 ∈ wcel 2137 ∃!weu 2605 {cab 2744 {csn 4319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-v 3340 df-sn 4320 |
This theorem is referenced by: initoid 16854 termoid 16855 initoeu2lem1 16863 funpartfv 32356 irinitoringc 42577 |
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