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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 5814 | . 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 2106 Vcvv 3478 {csn 4631 × cxp 5687 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-sn 4632 df-rel 5696 |
This theorem is referenced by: setscom 17214 setsid 17242 |
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