Theorem rabsn 2441

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 1531	. . . . 5 |- (x = B -> (x e. A <-> B e. A))
2	1	pm5.32ri 645	. . . 4 |- ((x e. A /\ x = B) <-> (B e. A /\ x = B))
3	2	baib 684	. . 3 |- (B e. A -> ((x e. A /\ x = B) <-> x = B))
4	3	abbidv 1574	. 2 |- (B e. A -> {x | (x e. A /\ x = B)} = {x | x = B})
5		df-rab 1649	. 2 |- {x e. A | x = B} = {x | (x e. A /\ x = B)}
6		df-sn 2408	. 2 |- {B} = {x | x = B}
7	4, 5, 6	3eqtr4g 1528	1 |- (B e. A -> {x e. A | x = B} = {B})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461  {crab 1645  {csn 2405

This theorem is referenced by:  unisn3 2871  pjspansnt 9440  iint 10514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-rab 1649  df-sn 2408

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