Theorem shex 10710

Description: The set of subspaces of a Hilbert space exists (is a set).

Assertion

Ref	Expression
shex	|- SH e. _V

Proof of Theorem shex

Step	Hyp	Ref	Expression
1		df-sh 10709	. 2 |- SH = {h | ((h C_ ~H /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))}
2		df-pw 3035	. . . 4 |- ~P~H = {h | h C_ ~H}
3		ax-hilex 10501	. . . . 5 |- ~H e. _V
4	3	pwex 3487	. . . 4 |- ~P~H e. _V
5	2, 4	eqeltrri 1968	. . 3 |- {h | h C_ ~H} e. _V
6		simpll 448	. . . 4 |- (((h C_ ~H /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h)) -> h C_ ~H)
7	6	ss2abi 2679	. . 3 |- {h | ((h C_ ~H /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))} C_ {h | h C_ ~H}
8	5, 7	ssexi 3456	. 2 |- {h | ((h C_ ~H /\ 0h e. h) /\ (A.x e. h A.y e. h (x +h y) e. h /\ A.x e. CC A.y e. h (x .h y) e. h))} e. _V
9	1, 8	eqeltri 1967	1 |- SH e. _V

Syntax hints:   /\ wa 240   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  (class class class)co 4884  CCcc 6384  ~Hchil 10420   +h cva 10421   .h csm 10422  0hc0v 10423  SHcsh 10429

This theorem is referenced by:  chex 10728

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-pow 3481  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-ss 2605  df-pw 3035  df-sh 10709

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