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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 4730. (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 4708 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) | |
2 | 1 | anidms 567 | . 2 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}}) |
3 | dfsn2 4485 | . . . . 5 ⊢ {𝐴} = {𝐴, 𝐴} | |
4 | 3 | eqcomi 2804 | . . . 4 ⊢ {𝐴, 𝐴} = {𝐴} |
5 | 4 | preq2i 4580 | . . 3 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
6 | dfsn2 4485 | . . 3 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 5, 6 | eqtr4i 2822 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}} |
8 | 2, 7 | syl6eq 2847 | 1 ⊢ (𝐴 ∈ 𝑉 → 〈𝐴, 𝐴〉 = {{𝐴}}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 {csn 4472 {cpr 4474 〈cop 4478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 |
This theorem is referenced by: opid 4730 brin3 35208 |
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