Theorem hlex 8530

Description: The base set of a Hilbert space is a set.

Hypothesis

Ref	Expression
hlex.1	|- X = (Base` U)

Assertion

Ref	Expression
hlex	|- X e. V

Proof of Theorem hlex

Step	Hyp	Ref	Expression
1		hlex.1	. 2 |- X = (Base` U)
2		fvex 3717	. 2 |- (Base` U) e. V
3	1, 2	eqeltr 1536	1 |- X e. V

Syntax hints:   = wceq 953   e. wcel 955  Vcvv 1802  ` cfv 3172  Basecba 8143

This theorem is referenced by:  htthlem3 8552  h2hcau 8788  h2hlm 8789

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494  df-fv 3188

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