Theorem relsn 4443

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 4352	. 2 |- ( Rel { A } <-> { A } C_ ( _V X. _V ) )
2		relsn.1	. . 3 |- A e. _V
3	2	snss 3494	. 2 |- ( A e. ( _V X. _V ) <-> { A } C_ ( _V X. _V ) )
4	1, 3	bitr4i 176	1 |- ( Rel { A } <-> A e. ( _V X. _V ) )

Syntax hints:   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375   X. cxp 4343   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-sn 3381  df-rel 4352

This theorem is referenced by:  relsnop  4444  relsn2m  4791