Theorem relsn 4946

Description: A singleton is a relation iff it is an ordered pair. (Contributed by NM, 24-Sep-2013.)

Hypothesis

Ref	Expression
relsn.1	|- A e. _V

Assertion

Ref	Expression
relsn	|- ( Rel { A } <-> A e. ( _V X. _V ) )

Proof of Theorem relsn

Step	Hyp	Ref	Expression
1		df-rel 4852	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		relsn.1	. . 3 |- A e. _V
3	2	snss 3894	. 2 |- ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) )
4	1, 3	bitr4i 244	1 |- ( Rel { A } <-> A e. ( _V X. _V ) )

Syntax hints:   <-> wb 177   e. wcel 1721   _Vcvv 2924   C_ wss 3288   {csn 3782   X. cxp 4843   Rel wrel 4850

This theorem is referenced by:  relsnop  4947  relsn2  5307  setscom  13460  setsid  13471

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-v 2926  df-in 3295  df-ss 3302  df-sn 3788  df-rel 4852