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Mirrors > Home > ILE Home > Th. List > snsstp1 | GIF version |
Description: A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) |
Ref | Expression |
---|---|
snsstp1 | ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snsspr1 3533 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
2 | ssun1 3135 | . . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sstri 3008 | . 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) |
4 | df-tp 3406 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
5 | 3, 4 | sseqtr4i 3032 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 2971 ⊆ wss 2973 {csn 3398 {cpr 3399 {ctp 3400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pr 3405 df-tp 3406 |
This theorem is referenced by: (None) |
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