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Mirrors > Home > ILE Home > Th. List > prsstp12 | GIF version |
Description: A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.) |
Ref | Expression |
---|---|
prsstp12 | ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 3239 | . 2 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | df-tp 3535 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sseqtrri 3132 | 1 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3069 ⊆ wss 3071 {csn 3527 {cpr 3528 {ctp 3529 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-tp 3535 |
This theorem is referenced by: prsstp13 3674 prsstp23 3675 sstpr 3684 |
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