|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > tpidm23 | Structured version Visualization version GIF version | ||
| Description: Unordered triple {𝐴, 𝐵, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.) | 
| Ref | Expression | 
|---|---|
| tpidm23 | ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | tprot 4749 | . 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐵, 𝐵, 𝐴} | |
| 2 | tpidm12 4755 | . 2 ⊢ {𝐵, 𝐵, 𝐴} = {𝐵, 𝐴} | |
| 3 | prcom 4732 | . 2 ⊢ {𝐵, 𝐴} = {𝐴, 𝐵} | |
| 4 | 1, 2, 3 | 3eqtri 2769 | 1 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 {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: tppreq3 4759 fntpb 7229 hashtpg 14524 hash3tpde 14532 | 
| Copyright terms: Public domain | W3C validator |