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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 3752 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
2 | ssun1 3310 | . . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sstri 3176 | . 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) |
4 | df-tp 3612 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
5 | 3, 4 | sseqtrri 3202 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3139 ⊆ wss 3141 {csn 3604 {cpr 3605 {ctp 3606 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pr 3611 df-tp 3612 |
This theorem is referenced by: cnfldbas 13672 |
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