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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 3166 | . 2 ⊢ {𝐴, 𝐵} ⊆ ({𝐴, 𝐵} ∪ {𝐶}) | |
2 | df-tp 3460 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 1, 2 | sseqtr4i 3062 | 1 ⊢ {𝐴, 𝐵} ⊆ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff set class |
Syntax hints: ∪ cun 3000 ⊆ wss 3002 {csn 3452 {cpr 3453 {ctp 3454 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-tp 3460 |
This theorem is referenced by: prsstp13 3599 prsstp23 3600 sstpr 3609 |
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