Theorem rabsn 3685

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 2256	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	pm5.32ri 455	. . . 4 |- ( ( x e. A /\ x = B ) <-> ( B e. A /\ x = B ) )
3	2	baib 920	. . 3 |- ( B e. A -> ( ( x e. A /\ x = B ) <-> x = B ) )
4	3	abbidv 2311	. 2 |- ( B e. A -> { x | ( x e. A /\ x = B ) } = { x | x = B } )
5		df-rab 2481	. 2 |- { x e. A | x = B } = { x | ( x e. A /\ x = B ) }
6		df-sn 3624	. 2 |- { B } = { x | x = B }
7	4, 5, 6	3eqtr4g 2251	1 |- ( B e. A -> { x e. A | x = B } = { B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   {cab 2179   {crab 2476   {csn 3618

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-rab 2481  df-sn 3624

This theorem is referenced by:  unisn3  4476