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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 4279 | . . 3 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵}) | |
2 | 1 | eqcoms 2630 | . 2 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵, 𝐵}) |
3 | tpidm23 4292 | . 2 ⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵} | |
4 | 2, 3 | syl6eq 2672 | 1 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 {cpr 4179 {ctp 4181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 df-tp 4182 |
This theorem is referenced by: tpprceq3 4335 1to3vfriswmgr 27144 |
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