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Mirrors > Home > ILE Home > Th. List > tpass | GIF version |
Description: Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
tpass | ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3578 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
2 | tprot 3663 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
3 | uncom 3261 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴}) | |
4 | 1, 2, 3 | 3eqtr4i 2195 | 1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∪ cun 3109 {csn 3570 {cpr 3571 {ctp 3572 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-tp 3578 |
This theorem is referenced by: qdassr 3668 |
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