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Mirrors > Home > MPE Home > Th. List > snsspr2 | Structured version Visualization version GIF version |
Description: A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) |
Ref | Expression |
---|---|
snsspr2 | ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 4131 | . 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 4587 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtrri 3979 | 1 ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3906 ⊆ wss 3908 {csn 4584 {cpr 4586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3445 df-un 3913 df-in 3915 df-ss 3925 df-pr 4587 |
This theorem is referenced by: snsstp2 4775 ord3ex 5340 ltrelxr 11212 2strop 17099 2strop1 17103 phlip 17224 prdsco 17342 ipotset 18414 gsumpr 19723 lsppratlem4 20596 ex-res 29271 subfacp1lem2a 33643 dvh3dim3N 39879 algvsca 41447 corclrcl 41921 mnuprdlem4 42497 |
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