Theorem unisn3 4493

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 3700	. . 3 |- ( A e. B -> { x e. B | x = A } = { A } )
2	1	unieqd 3861	. 2 |- ( A e. B -> U. { x e. B | x = A } = U. { A } )
3		unisng 3867	. 2 |- ( A e. B -> U. { A } = A )
4	2, 3	eqtrd 2238	1 |- ( A e. B -> U. { x e. B | x = A } = A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   {crab 2488   {csn 3633   U.cuni 3850

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-rab 2493  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851

This theorem is referenced by: (None)