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| 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 4631 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
| 2 | tprot 4749 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
| 3 | uncom 4158 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴}) | |
| 4 | 1, 2, 3 | 3eqtr4i 2775 | 1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: qdassr 4754 en3 9316 wuntp 10751 symgvalstruct 19414 symgvalstructOLD 19415 ex-pw 30448 |
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