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Mirrors > Home > ILE Home > Th. List > opid | GIF version |
Description: The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) |
Ref | Expression |
---|---|
opid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
opid | ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3412 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | eqcomi 2085 | . . 3 ⊢ {𝐴, 𝐴} = {𝐴} |
3 | 2 | preq2i 3473 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
4 | opid.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4, 4 | dfop 3569 | . 2 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}} |
6 | dfsn2 3412 | . 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 3, 5, 6 | 3eqtr4i 2111 | 1 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 Vcvv 2601 {csn 3398 {cpr 3399 〈cop 3401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: dmsnsnsng 4818 funopg 4954 |
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