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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 3614 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
2 | tprot 3699 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
3 | uncom 3293 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴}) | |
4 | 1, 2, 3 | 3eqtr4i 2219 | 1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∪ cun 3141 {csn 3606 {cpr 3607 {ctp 3608 |
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 2170 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-v 2753 df-un 3147 df-sn 3612 df-pr 3613 df-tp 3614 |
This theorem is referenced by: qdassr 3704 |
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