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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 4261 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥}) | |
2 | abid2 2745 | . . . 4 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | eqeq1i 2627 | . . 3 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} = {𝑥} ↔ 𝐴 = {𝑥}) |
4 | 3 | exbii 1774 | . 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 1483 ∃wex 1704 ∈ wcel 1990 ∃!weu 2470 {cab 2608 {csn 4177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-sn 4178 |
This theorem is referenced by: initoid 16655 termoid 16656 initoeu2lem1 16664 funpartfv 32052 irinitoringc 42069 |
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