![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tpass | Structured version Visualization version 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 4635 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
2 | tprot 4756 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
3 | uncom 4152 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴}) | |
4 | 1, 2, 3 | 3eqtr4i 2765 | 1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3945 {csn 4630 {cpr 4632 {ctp 4634 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-v 3473 df-un 3952 df-sn 4631 df-pr 4633 df-tp 4635 |
This theorem is referenced by: qdassr 4761 en3 9311 wuntp 10740 symgvalstruct 19356 symgvalstructOLD 19357 ex-pw 30257 |
Copyright terms: Public domain | W3C validator |