Theorem relsn2m 5114

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 4746	. 2 |- ( Rel { A } <-> A e. ( _V X. _V ) )
3		dmsnm 5109	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
4	2, 3	bitri 184	1 |- ( Rel { A } <-> E. x x e. dom { A } )

Syntax hints:   <-> wb 105   E.wex 1503   e. wcel 2160   _Vcvv 2752   {csn 3607   X. cxp 4639   dom cdm 4641   Rel wrel 4646

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4647  df-rel 4648  df-dm 4651

This theorem is referenced by: (None)