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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 5797 | . 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 205 ∈ wcel 2099 Vcvv 3470 {csn 4624 × cxp 5670 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-ss 3962 df-sn 4625 df-rel 5679 |
This theorem is referenced by: setscom 17142 setsid 17170 |
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