Theorem rabsn 3700

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 2268	. . . . 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 921	. . 3 |- ( B e. A -> ( ( x e. A /\ x = B ) <-> x = B ) )
4	3	abbidv 2323	. 2 |- ( B e. A -> { x | ( x e. A /\ x = B ) } = { x | x = B } )
5		df-rab 2493	. 2 |- { x e. A | x = B } = { x | ( x e. A /\ x = B ) }
6		df-sn 3639	. 2 |- { B } = { x | x = B }
7	4, 5, 6	3eqtr4g 2263	1 |- ( B e. A -> { x e. A | x = B } = { B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {cab 2191   {crab 2488   {csn 3633

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-rab 2493  df-sn 3639

This theorem is referenced by:  unisn3  4492