Theorem rabsn 3094

Description: Condition where a restricted class abstraction is a singleton.

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 1957	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
2	1	pm5.32ri 708	. . . 4 |- ((x e. A /\ x = B) <-> (B e. A /\ x = B))
3	2	baib 749	. . 3 |- (B e. A -> ((x e. A /\ x = B) <-> x = B))
4	3	abbidv 2008	. 2 |- (B e. A -> {x | (x e. A /\ x = B)} = {x | x = B})
5		df-rab 2112	. 2 |- {x e. A | x = B} = {x | (x e. A /\ x = B)}
6		df-sn 3049	. 2 |- {B} = {x | x = B}
7	4, 5, 6	3eqtr4g 1953	1 |- (B e. A -> {x e. A | x = B} = {B})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  {crab 2108  {csn 3044

This theorem is referenced by:  unisn3 3800  pjspansn 11133  iint 15012  pmapat 17243

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rab 2112  df-sn 3049

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