Theorem unisn3 2882

Description: Union of a singleton in the form of a restricted class abstraction.

Assertion

Ref	Expression
unisn3	|- (A e. B -> U.{x e. B | x = A} = A)

Distinct variable groups:   x,A   x,B

Proof of Theorem unisn3

Step	Hyp	Ref	Expression
1		rabsn 2449	. . 3 |- (A e. B -> {x e. B | x = A} = {A})
2	1	unieqd 2516	. 2 |- (A e. B -> U.{x e. B | x = A} = U.{A})
3		unisng 2522	. 2 |- (A e. B -> U.{A} = A)
4	2, 3	eqtrd 1510	1 |- (A e. B -> U.{x e. B | x = A} = A)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  {crab 1651  {csn 2413  U.cuni 2507

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-uni 2508

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