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Mirrors > Home > MPE Home > Th. List > snsstp3 | Structured version Visualization version GIF version |
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
Ref | Expression |
---|---|
snsstp3 | ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3739 | . 2 ⊢ {𝐶} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | df-tp 4130 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sseqtr4i 3601 | 1 ⊢ {𝐶} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3538 ⊆ wss 3540 {csn 4125 {cpr 4127 {ctp 4129 |
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-un 3545 df-in 3547 df-ss 3554 df-tp 4130 |
This theorem is referenced by: fr3nr 6871 rngmulr 15826 srngmulr 15834 lmodsca 15843 ipsmulr 15850 ipsip 15853 phlsca 15860 topgrptset 15868 otpsle 15877 otpsleOLD 15881 odrngmulr 15892 odrngds 15895 prdsmulr 15942 prdsip 15944 prdsds 15947 imasds 15996 imasmulr 16001 imasip 16004 fuccofval 16442 setccofval 16555 catccofval 16573 estrccofval 16592 xpccofval 16645 psrmulr 19205 cnfldmul 19573 cnfldds 19577 trkgitv 25146 signswch 29964 algmulr 36769 clsk1indlem1 37363 rngccofvalALTV 41779 ringccofvalALTV 41842 |
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