Theorem relsn2m 5154

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 4781	. 2 |- ( Rel { A } <-> A e. ( _V X. _V ) )
3		dmsnm 5149	. 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 1515   e. wcel 2176   _Vcvv 2772   {csn 3633   X. cxp 4674   dom cdm 4676   Rel wrel 4681

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-dm 4686

This theorem is referenced by: (None)