Theorem rabsn 3459

Description: Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Assertion

Ref	Expression
rabsn	|- ( B e. A -> { x e. A | x = B } = { B } )

Distinct variable groups:   x, A   x, B

Proof of Theorem rabsn

Step	Hyp	Ref	Expression
1		eleq1 2141	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	pm5.32ri 442	. . . 4 |- ( ( x e. A /\ x = B ) <-> ( B e. A /\ x = B ) )
3	2	baib 861	. . 3 |- ( B e. A -> ( ( x e. A /\ x = B ) <-> x = B ) )
4	3	abbidv 2196	. 2 |- ( B e. A -> { x | ( x e. A /\ x = B ) } = { x | x = B } )
5		df-rab 2357	. 2 |- { x e. A | x = B } = { x | ( x e. A /\ x = B ) }
6		df-sn 3404	. 2 |- { B } = { x | x = B }
7	4, 5, 6	3eqtr4g 2138	1 |- ( B e. A -> { x e. A | x = B } = { B } )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   {crab 2352   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-rab 2357  df-sn 3404

This theorem is referenced by:  unisn3  4198