Theorem relsn2m 5081

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 4716	. 2 |- ( Rel { A } <-> A e. ( _V X. _V ) )
3		dmsnm 5076	. 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 1485   e. wcel 2141   _Vcvv 2730   {csn 3583   X. cxp 4609   dom cdm 4611   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621

This theorem is referenced by: (None)