Theorem relsng 4762

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	|- ( A e. V -> ( Rel { A } <-> A e. ( _V X. _V ) ) )

Proof of Theorem relsng

Step	Hyp	Ref	Expression
1		df-rel 4666	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		snssg 3752	. 2 |- ( A e. V -> ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) ) )
3	1, 2	bitr4id 199	1 |- ( A e. V -> ( Rel { A } <-> A e. ( _V X. _V ) ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {csn 3618   X. cxp 4657   Rel wrel 4664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-rel 4666

This theorem is referenced by:  relsnopg  4763  setscom  12658  setsslid  12669