Theorem relsn 5014

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 4920	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		relsn.1	. . 3 |- A e. _V
3	2	snss 3955	. 2 |- ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) )
4	1, 3	bitr4i 245	1 |- ( Rel { A } <-> A e. ( _V X. _V ) )

Syntax hints:   <-> wb 178   e. wcel 1728   _Vcvv 2965   C_ wss 3309   {csn 3843   X. cxp 4911   Rel wrel 4918

This theorem is referenced by:  relsnop  5015  relsn2  5375  setscom  13535  setsid  13546

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424

This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2967  df-in 3316  df-ss 3323  df-sn 3849  df-rel 4920