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| Mirrors > Home > MPE Home > Th. List > snsstp1 | 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 |
|---|---|
| snsstp1 | ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snsspr1 4814 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
| 2 | ssun1 4178 | . . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
| 3 | 1, 2 | sstri 3993 | . 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) |
| 4 | df-tp 4631 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
| 5 | 3, 4 | sseqtrri 4033 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-ss 3968 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: fr3nr 7792 rngbase 17343 srngbase 17354 lmodbase 17370 ipsbase 17381 ipssca 17384 phlbase 17391 topgrpbas 17406 otpsbas 17421 odrngbas 17448 odrngtset 17451 prdssca 17501 prdsbas 17502 prdstset 17511 imasbas 17557 imassca 17564 imastset 17567 fucbas 18008 setcbas 18123 catcbas 18146 estrcbas 18169 cnfldbas 21368 cnfldtset 21374 cnfldbasOLD 21383 cnfldtsetOLD 21387 psrbas 21953 psrsca 21967 trkgbas 28453 rlocbas 33271 rlocaddval 33272 rlocmulval 33273 idlsrgbas 33532 signswch 34576 algbase 43186 clsk1indlem4 44057 clsk1indlem1 44058 cycl3grtri 47914 rngcbasALTV 48182 ringcbasALTV 48216 mndtcbasval 49177 |
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