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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 4572 | . 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
2 | tprot 4685 | . 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
3 | uncom 4129 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴}) | |
4 | 1, 2, 3 | 3eqtr4i 2854 | 1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3934 {csn 4567 {cpr 4569 {ctp 4571 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3941 df-sn 4568 df-pr 4570 df-tp 4572 |
This theorem is referenced by: qdassr 4690 en3 8755 wuntp 10133 symgvalstruct 18525 ex-pw 28208 |
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