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| Mirrors > Home > MPE Home > Th. List > snsspr1 | Structured version Visualization version GIF version | ||
| Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.) |
| Ref | Expression |
|---|---|
| snsspr1 | ⊢ {𝐴} ⊆ {𝐴, 𝐵} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssun1 4178 | . 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵}) | |
| 2 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 3 | 1, 2 | sseqtrri 4033 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵} |
| Colors of variables: wff setvar class |
| Syntax hints: ∪ cun 3949 ⊆ wss 3951 {csn 4626 {cpr 4628 |
| 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 |
| This theorem is referenced by: snsstp1 4816 op1stb 5476 uniop 5520 1sdom2dom 9283 rankopb 9892 ltrelxr 11322 seqexw 14058 2strbas 17268 2strbas1 17272 phlvsca 17394 prdshom 17512 ipobas 18576 ipolerval 18577 gsumpr 19973 lspprid1 20995 lsppratlem3 21151 lsppratlem4 21152 ex-dif 30442 ex-un 30443 ex-in 30444 idlsrgtset 33536 coinflippv 34486 pthhashvtx 35133 subfacp1lem2a 35185 altopthsn 35962 rankaltopb 35980 dvh3dim3N 41451 mapdindp2 41723 lspindp5 41772 algsca 43189 clsk1indlem2 44055 clsk1indlem3 44056 clsk1indlem1 44058 mnuprdlem4 44294 |
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