Theorem relsng 4730

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 4634	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		snssg 3727	. 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 2148   _Vcvv 2738   C_ wss 3130   {csn 3593   X. cxp 4625   Rel wrel 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-sn 3599  df-rel 4634

This theorem is referenced by:  relsnopg  4731  setscom  12502  setsslid  12513