Theorem relsn2m 5009

Description: A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

Hypothesis

Ref	Expression
relsn2m.1	|- A e. _V

Assertion

Ref	Expression
relsn2m	|- ( Rel { A } <-> E. x x e. dom { A } )

Distinct variable group:   x, A

Proof of Theorem relsn2m

Step	Hyp	Ref	Expression
1		relsn2m.1	. . 3 |- A e. _V
2	1	relsn 4644	. 2 |- ( Rel { A } <-> A e. ( _V X. _V ) )
3		dmsnm 5004	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
4	2, 3	bitri 183	1 |- ( Rel { A } <-> E. x x e. dom { A } )

Syntax hints:   <-> wb 104   E.wex 1468   e. wcel 1480   _Vcvv 2686   {csn 3527   X. cxp 4537   dom cdm 4539   Rel wrel 4544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-dm 4549

This theorem is referenced by: (None)