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Mirrors > Home > MPE Home > Th. List > opidg | Structured version Visualization version GIF version |
Description: The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Closed form of opid 4776. (Contributed by Peter Mazsa, 22-Jul-2019.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
opidg | ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 4752 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) | |
2 | 1 | anidms 571 | . 2 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) |
3 | dfsn2 4528 | . . . . 5 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 3 | eqcomi 2768 | . . . 4 ⊢ {𝐴, 𝐴} = {𝐴} |
5 | 4 | preq2i 4623 | . . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
6 | dfsn2 4528 | . . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 5, 6 | eqtr4i 2785 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}} |
8 | 2, 7 | eqtrdi 2810 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2112 {csn 4515 {cpr 4517 〈cop 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-dif 3857 df-un 3859 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 |
This theorem is referenced by: opid 4776 brin3 36083 |
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