|   | 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 4179 | . 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵}) | |
| 2 | df-pr 4629 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
| 3 | 1, 2 | sseqtrri 4033 | 1 ⊢ {𝐵} ⊆ {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: snsstp2 4817 ord3ex 5387 ltrelxr 11322 2strop 17269 2strop1 17273 phlip 17395 prdsco 17513 ipotset 18578 gsumpr 19973 lsppratlem4 21152 ex-res 30460 subfacp1lem2a 35185 dvh3dim3N 41451 algvsca 43190 corclrcl 43720 mnuprdlem4 44294 | 
| Copyright terms: Public domain | W3C validator |