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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 4147 | . 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 4562 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtrri 4003 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3933 ⊆ wss 3935 {csn 4559 {cpr 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 2793 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-un 3940 df-in 3942 df-ss 3951 df-pr 4562 |
This theorem is referenced by: snsstp1 4743 op1stb 5355 uniop 5397 rankopb 9270 ltrelxr 10691 seqexw 13375 2strbas 16593 2strbas1 16596 phlvsca 16647 prdshom 16730 ipobas 17755 ipolerval 17756 gsumpr 19006 lspprid1 19700 lsppratlem3 19852 lsppratlem4 19853 ex-dif 28130 ex-un 28131 ex-in 28132 coinflippv 31641 pthhashvtx 32272 subfacp1lem2a 32325 altopthsn 33320 rankaltopb 33338 dvh3dim3N 38467 mapdindp2 38739 lspindp5 38788 algsca 39661 clsk1indlem2 40272 clsk1indlem3 40273 clsk1indlem1 40275 mnuprdlem4 40491 |
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