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Mirrors > Home > MPE Home > Th. List > opid | Structured version Visualization version GIF version |
Description: The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. Inference form of opidg 4824. (Contributed by FL, 28-Dec-2011.) (Proof shortened by AV, 16-Feb-2022.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
opid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
opid | ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opid.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opidg 4824 | . 2 ⊢ (𝐴 ∈ V → 〈𝐴, 𝐴〉 = {{𝐴}}) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 |
This theorem is referenced by: dmsnsnsn 6079 funopg 6391 vtxval3sn 26830 iedgval3sn 26831 |
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