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Theorem relsng 4610
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 3624 . 2 (𝐴𝑉 → (𝐴 ∈ (V × V) ↔ {𝐴} ⊆ (V × V)))
2 df-rel 4514 . 2 (Rel {𝐴} ↔ {𝐴} ⊆ (V × V))
31, 2syl6rbbr 198 1 (𝐴𝑉 → (Rel {𝐴} ↔ 𝐴 ∈ (V × V)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1463  Vcvv 2658  wss 3039  {csn 3495   × cxp 4505  Rel wrel 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-in 3045  df-ss 3052  df-sn 3501  df-rel 4514
This theorem is referenced by:  relsnopg  4611  setscom  11905  setsslid  11915
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