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Mirrors > Home > MPE Home > Th. List > elsng | Structured version Visualization version GIF version |
Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
elsng | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2614 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
2 | df-sn 4126 | . 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵} | |
3 | 1, 2 | elab2g 3322 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 |
This theorem is referenced by: elsn 4140 elsni 4142 snidg 4153 nelsnOLD 4160 eltpg 4174 eldifsn 4260 sneqrg 4310 elsucg 5709 ltxr 11825 elfzp12 12288 fprodn0f 14561 lcmfunsnlem2 15191 ramcl 15571 initoeu2lem1 16487 pmtrdifellem4 17722 logbmpt 24326 2lgslem2 24920 nbcusgra 25992 frgrancvvdeqlem1 26557 xrge0tsmsbi 29117 elzrhunit 29351 elzdif0 29352 esumrnmpt2 29457 plymulx 29951 bj-projval 32177 reclimc 38720 itgsincmulx 38866 dirkercncflem2 38997 dirkercncflem4 38999 fourierdlem53 39052 fourierdlem58 39057 fourierdlem60 39059 fourierdlem61 39060 fourierdlem62 39061 fourierdlem76 39075 fourierdlem101 39100 elaa2 39127 etransc 39176 qndenserrnbl 39191 sge0tsms 39273 el1fzopredsuc 39948 frgrncvvdeqlem1 41469 |
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