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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 3919 | . 2 ⊢ {𝐴} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 4324 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtr4i 3779 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3713 ⊆ wss 3715 {csn 4321 {cpr 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-un 3720 df-in 3722 df-ss 3729 df-pr 4324 |
This theorem is referenced by: snsstp1 4492 op1stb 5088 uniop 5125 rankopb 8888 ltrelxr 10291 2strbas 16186 2strbas1 16189 phlvsca 16240 prdshom 16329 ipobas 17356 ipolerval 17357 lspprid1 19199 lsppratlem3 19351 lsppratlem4 19352 ex-dif 27591 ex-un 27592 ex-in 27593 coinflippv 30854 subfacp1lem2a 31469 altopthsn 32374 rankaltopb 32392 dvh3dim3N 37240 mapdindp2 37512 lspindp5 37561 algsca 38253 clsk1indlem2 38842 clsk1indlem3 38843 clsk1indlem1 38845 gsumpr 42649 |
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