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| Mirrors > Home > MPE Home > Th. List > relsn | Structured version Visualization version GIF version | ||
| Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) | 
| Ref | Expression | 
|---|---|
| relsn.1 | ⊢ 𝐴 ∈ V | 
| Ref | Expression | 
|---|---|
| relsn | ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relsn.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | relsng 5811 | . 2 ⊢ (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel {𝐴} ↔ 𝐴 ∈ (V × V)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 Vcvv 3480 {csn 4626 × cxp 5683 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-sn 4627 df-rel 5692 | 
| This theorem is referenced by: setscom 17217 setsid 17244 | 
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