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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 3389 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | eqcomi 2044 | . . 3 ⊢ {𝐴, 𝐴} = {𝐴} |
3 | 2 | preq2i 3451 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
4 | opid.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4, 4 | dfop 3548 | . 2 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}} |
6 | dfsn2 3389 | . 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 3, 5, 6 | 3eqtr4i 2070 | 1 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 Vcvv 2557 {csn 3375 {cpr 3376 〈cop 3378 |
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-3an 887 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-sn 3381 df-pr 3382 df-op 3384 |
This theorem is referenced by: dmsnsnsng 4798 funopg 4934 |
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