Theorem unisn3 4180

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 3437	. . 3 |- ( A e. B -> { x e. B | x = A } = { A } )
2	1	unieqd 3591	. 2 |- ( A e. B -> U. { x e. B | x = A } = U. { A } )
3		unisng 3597	. 2 |- ( A e. B -> U. { A } = A )
4	2, 3	eqtrd 2072	1 |- ( A e. B -> U. { x e. B | x = A } = A )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   {crab 2310   {csn 3375   U.cuni 3580

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581

This theorem is referenced by: (None)