Theorem relsng 4766

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 4670	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		snssg 3756	. 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 2167   _Vcvv 2763   C_ wss 3157   {csn 3622   X. cxp 4661   Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-sn 3628  df-rel 4670

This theorem is referenced by:  relsnopg  4767  setscom  12718  setsslid  12729