Theorem relsng 4707

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 4611	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		snssg 3709	. 2 |- ( A e. V -> ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) ) )
3	1, 2	bitr4id 198	1 |- ( A e. V -> ( Rel { A } <-> A e. ( _V X. _V ) ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   _Vcvv 2726   C_ wss 3116   {csn 3576   X. cxp 4602   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-sn 3582  df-rel 4611

This theorem is referenced by:  relsnopg  4708  setscom  12434  setsslid  12444