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Theorem relsn 5383
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 5381 . 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 196  wcel 2140  Vcvv 3341  {csn 4322   × cxp 5265  Rel wrel 5272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-v 3343  df-in 3723  df-ss 3730  df-sn 4323  df-rel 5274
This theorem is referenced by:  relsnopOLD  5386  relsn2OLD  5765  setscom  16126  setsid  16137
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