MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  relsng Structured version   Visualization version   GIF version

Theorem relsng 5428
Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.) (Revised by BJ, 12-Feb-2022.)
Assertion
Ref Expression
relsng (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))

Proof of Theorem relsng
StepHypRef Expression
1 snssg 4504 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
2 df-rel 5320 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
31, 2syl6rbbr 282 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  Vcvv 3386  wss 3770  {csn 4369   × cxp 5311  Rel wrel 5318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-v 3388  df-in 3777  df-ss 3784  df-sn 4370  df-rel 5320
This theorem is referenced by:  relsnopg  5429  relsn  5430  relsn2  5822
  Copyright terms: Public domain W3C validator