Theorem unisn3 4510

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 3710	. . 3 |- ( A e. B -> { x e. B | x = A } = { A } )
2	1	unieqd 3875	. 2 |- ( A e. B -> U. { x e. B | x = A } = U. { A } )
3		unisng 3881	. 2 |- ( A e. B -> U. { A } = A )
4	2, 3	eqtrd 2240	1 |- ( A e. B -> U. { x e. B | x = A } = A )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2178   {crab 2490   {csn 3643   U.cuni 3864

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-rab 2495  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-uni 3865

This theorem is referenced by: (None)