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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 5667 | . 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 207 ∈ wcel 2105 Vcvv 3492 {csn 4557 × cxp 5546 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-in 3940 df-ss 3949 df-sn 4558 df-rel 5555 |
This theorem is referenced by: setscom 16515 setsid 16526 |
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