Theorem rabsn 3775

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 2301	. . . . 5 |- ( x = B -> ( x e. A <-> B e. A ) )
2	1	pm5.32ri 459	. . . 4 |- ( ( x e. A /\ x = B ) <-> ( B e. A /\ x = B ) )
3	2	baib 931	. . 3 |- ( B e. A -> ( ( x e. A /\ x = B ) <-> x = B ) )
4	3	abbidv 2358	. 2 |- ( B e. A -> { x | ( x e. A /\ x = B ) } = { x | x = B } )
5		df-rab 2537	. 2 |- { x e. A | x = B } = { x | ( x e. A /\ x = B ) }
6		df-sn 3714	. 2 |- { B } = { x | x = B }
7	4, 5, 6	3eqtr4g 2296	1 |- ( B e. A -> { x e. A | x = B } = { B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   {csn 3708

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rab 2537  df-sn 3714

This theorem is referenced by:  unisn3  4589  ssenneg  11261