Theorem rabsn 3650

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 2233	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	pm5.32ri 452	. . . 4 |- ( ( x e. A /\ x = B ) <-> ( B e. A /\ x = B ) )
3	2	baib 914	. . 3 |- ( B e. A -> ( ( x e. A /\ x = B ) <-> x = B ) )
4	3	abbidv 2288	. 2 |- ( B e. A -> { x | ( x e. A /\ x = B ) } = { x | x = B } )
5		df-rab 2457	. 2 |- { x e. A | x = B } = { x | ( x e. A /\ x = B ) }
6		df-sn 3589	. 2 |- { B } = { x | x = B }
7	4, 5, 6	3eqtr4g 2228	1 |- ( B e. A -> { x e. A | x = B } = { B } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   {cab 2156   {crab 2452   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-rab 2457  df-sn 3589

This theorem is referenced by:  unisn3  4430