Theorem unisn3 4480

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 3689	. . 3 |- ( A e. B -> { x e. B | x = A } = { A } )
2	1	unieqd 3850	. 2 |- ( A e. B -> U. { x e. B | x = A } = U. { A } )
3		unisng 3856	. 2 |- ( A e. B -> U. { A } = A )
4	2, 3	eqtrd 2229	1 |- ( A e. B -> U. { x e. B | x = A } = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {crab 2479   {csn 3622   U.cuni 3839

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-uni 3840

This theorem is referenced by: (None)