Theorem relsn 4790

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 4696	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		relsn.1	. . 3 |- A e. _V
3	2	snss 3748	. 2 |- ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) )
4	1, 3	bitr4i 243	1 |- ( Rel { A } <-> A e. ( _V X. _V ) )

Syntax hints:   <-> wb 176   e. wcel 1684   _Vcvv 2788   C_ wss 3152   {csn 3640   X. cxp 4687   Rel wrel 4694

This theorem is referenced by:  relsnop  4791  relsn2  5143  setscom  13176  setsid  13187

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-in 3159  df-ss 3166  df-sn 3646  df-rel 4696