Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4087 | . 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 4544 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtrri 3938 | 1 ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3864 ⊆ wss 3866 {csn 4541 {cpr 4543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-un 3871 df-in 3873 df-ss 3883 df-pr 4544 |
This theorem is referenced by: snsstp2 4730 ord3ex 5280 ltrelxr 10894 2strop 16780 2strop1 16783 phlip 16884 prdsco 16973 ipotset 18039 gsumpr 19340 lsppratlem4 20187 ex-res 28524 subfacp1lem2a 32855 dvh3dim3N 39200 algvsca 40710 corclrcl 40992 mnuprdlem4 41566 |
Copyright terms: Public domain | W3C validator |