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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 4898. (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 4876 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) | |
2 | 1 | anidms 565 | . 2 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) |
3 | dfsn2 4645 | . . . . 5 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 3 | eqcomi 2737 | . . . 4 ⊢ {𝐴, 𝐴} = {𝐴} |
5 | 4 | preq2i 4746 | . . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
6 | dfsn2 4645 | . . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 5, 6 | eqtr4i 2759 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}} |
8 | 2, 7 | eqtrdi 2784 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4632 {cpr 4634 〈cop 4638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 |
This theorem is referenced by: opid 4898 brin3 37914 |
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