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Theorem relsn 5130
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 5030 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
2 relsn.1 . . 3 𝐴 ∈ V
32snss 4253 . 2 (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V))
41, 3bitr4i 265 1 (Rel {𝐴} ↔ 𝐴 ∈ (V × V))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wcel 1975  Vcvv 3167  wss 3534  {csn 4119   × cxp 5021  Rel wrel 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-v 3169  df-in 3541  df-ss 3548  df-sn 4120  df-rel 5030
This theorem is referenced by:  relsnop  5131  relsn2  5504  setscom  15672  setsid  15683
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