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Mirrors > Home > MPE Home > Th. List > tppreq3 | Structured version Visualization version GIF version |
Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.) |
Ref | Expression |
---|---|
tppreq3 | ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpeq3 4750 | . . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵}) | |
2 | 1 | eqcoms 2733 | . 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵}) |
3 | tpidm23 4763 | . 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵} | |
4 | 2, 3 | eqtrdi 2781 | 1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {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 2696 |
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 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-un 3949 df-sn 4631 df-pr 4633 df-tp 4635 |
This theorem is referenced by: tpprceq3 4809 1to3vfriswmgr 30162 |
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