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Theorem relsn 5146
 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 df-rel 5045 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 relsn.1 . . 3 𝐴 ∈ V
32snss 4259 . 2 (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
41, 3bitr4i 266 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {csn 4125   × cxp 5036  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-sn 4126  df-rel 5045 This theorem is referenced by:  relsnop  5147  relsn2  5523  setscom  15731  setsid  15742
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