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Theorem relsn 5670
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)
Hypothesis
Ref Expression
relsn.1 𝐴 ∈ V
Assertion
Ref Expression
relsn (Rel {𝐴} ↔ 𝐴 ∈ (V × V))

Proof of Theorem relsn
StepHypRef Expression
1 relsn.1 . 2 𝐴 ∈ V
2 relsng 5667 . 2 (𝐴 ∈ V → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
31, 2ax-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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