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Mirrors > Home > ILE Home > Th. List > snsspr2 | 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 3210 | . 2 ⊢ {𝐵} ⊆ ({𝐴} ∪ {𝐵}) | |
2 | df-pr 3504 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 1, 2 | sseqtrri 3102 | 1 ⊢ {𝐵} ⊆ {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3039 ⊆ wss 3041 {csn 3497 {cpr 3498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pr 3504 |
This theorem is referenced by: snsstp2 3641 ssprr 3653 ord3ex 4084 ltrelxr 7793 |
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