Theorem unisn3 4423

Description: Union of a singleton in the form of a restricted class abstraction. (Contributed by NM, 3-Jul-2008.)

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 3643	. . 3 |- ( A e. B -> { x e. B | x = A } = { A } )
2	1	unieqd 3800	. 2 |- ( A e. B -> U. { x e. B | x = A } = U. { A } )
3		unisng 3806	. 2 |- ( A e. B -> U. { A } = A )
4	2, 3	eqtrd 2198	1 |- ( A e. B -> U. { x e. B | x = A } = A )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   {crab 2448   {csn 3576   U.cuni 3789

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-rab 2453  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790

This theorem is referenced by: (None)