Theorem relsng 4714

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 4618	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		snssg 3716	. 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 2141   _Vcvv 2730   C_ wss 3121   {csn 3583   X. cxp 4609   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-in 3127  df-ss 3134  df-sn 3589  df-rel 4618

This theorem is referenced by:  relsnopg  4715  setscom  12456  setsslid  12466