Theorem relsn 4837

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 4738	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		relsn.1	. . 3 |- A e. _V
3	2	snss 3813	. 2 |- ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) )
4	1, 3	bitr4i 187	1 |- ( Rel { A } <-> A e. ( _V X. _V ) )

Syntax hints:   <-> wb 105   e. wcel 2202   _Vcvv 2803   C_ wss 3201   {csn 3673   X. cxp 4729   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-in 3207  df-ss 3214  df-sn 3679  df-rel 4738

This theorem is referenced by:  relsnop  4838  relsn2m  5214