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Mirrors > Home > MPE Home > Th. List > snsstp2 | 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 |
---|---|
snsstp2 | ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snsspr2 4740 | . . 3 ⊢ {𝐵} ⊆ {𝐴, 𝐵} | |
2 | ssun1 4145 | . . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sstri 3973 | . 2 ⊢ {𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) |
4 | df-tp 4562 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
5 | 3, 4 | sseqtrri 4001 | 1 ⊢ {𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3931 ⊆ wss 3933 {csn 4557 {cpr 4559 {ctp 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 df-in 3940 df-ss 3949 df-pr 4560 df-tp 4562 |
This theorem is referenced by: fr3nr 7483 rngplusg 16609 srngplusg 16617 lmodplusg 16626 ipsaddg 16633 ipsvsca 16636 phlplusg 16643 topgrpplusg 16651 otpstset 16658 odrngplusg 16669 odrngle 16672 prdsplusg 16719 prdsvsca 16721 prdsle 16723 imasplusg 16778 imasvsca 16781 imasle 16784 fuchom 17219 setchomfval 17327 catchomfval 17346 estrchomfval 17364 xpchomfval 17417 psrplusg 20089 psrvscafval 20098 cnfldadd 20478 cnfldle 20482 trkgdist 26159 algaddg 39657 clsk1indlem4 40272 rngchomfvalALTV 44183 ringchomfvalALTV 44246 |
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