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Mirrors > Home > ILE Home > Th. List > tpidm12 | GIF version |
Description: Unordered triple {𝐴, 𝐴, 𝐵} is just an overlong way to write {𝐴, 𝐵}. (Contributed by David A. Wheeler, 10-May-2015.) |
Ref | Expression |
---|---|
tpidm12 | ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3389 | . . 3 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | uneq1i 3093 | . 2 ⊢ ({𝐴} ∪ {𝐵}) = ({𝐴, 𝐴} ∪ {𝐵}) |
3 | df-pr 3382 | . 2 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
4 | df-tp 3383 | . 2 ⊢ {𝐴, 𝐴, 𝐵} = ({𝐴, 𝐴} ∪ {𝐵}) | |
5 | 2, 3, 4 | 3eqtr4ri 2071 | 1 ⊢ {𝐴, 𝐴, 𝐵} = {𝐴, 𝐵} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∪ cun 2915 {csn 3375 {cpr 3376 {ctp 3377 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-pr 3382 df-tp 3383 |
This theorem is referenced by: tpidm13 3470 tpidm23 3471 tpidm 3472 |
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