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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 3755 | . . 3 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
2 | ssun1 3313 | . . 3 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sstri 3179 | . 2 ⊢ {𝐴} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) |
4 | df-tp 3615 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
5 | 3, 4 | sseqtrri 3205 | 1 ⊢ {𝐴} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3142 ⊆ wss 3144 {csn 3607 {cpr 3608 {ctp 3609 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pr 3614 df-tp 3615 |
This theorem is referenced by: cnfldbas 13829 |
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