Theorem hlex 8596

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 3738	. 2 |- (Base` U) e. V
3	1, 2	eqeltr 1547	1 |- X e. V

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  ` cfv 3188  Basecba 8201

This theorem is referenced by:  htthlem3 8618  h2hcau 8844  h2hlm 8845  axhilex 8846

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-uni 2508  df-fv 3204

Copyright terms: Public domain