Theorem hon0 11356

Description: A Hilbert space operator is not empty.

Assertion

Ref	Expression
hon0	|- (T:~H-->~H -> -. T = (/))

Proof of Theorem hon0

Step	Hyp	Ref	Expression
1		ax-hv0cl 10505	. . 3 |- 0h e. ~H
2		n0i 2880	. . 3 |- (0h e. ~H -> -. ~H = (/))
3	1, 2	ax-mp 7	. 2 |- -. ~H = (/)
4		ffn 4562	. . . 4 |- (T:~H-->~H -> T Fn ~H)
5		fndmu 4514	. . . . 5 |- ((T Fn ~H /\ T Fn (/)) -> ~H = (/))
6	5	ex 402	. . . 4 |- (T Fn ~H -> (T Fn (/) -> ~H = (/)))
7	4, 6	syl 12	. . 3 |- (T:~H-->~H -> (T Fn (/) -> ~H = (/)))
8		fn0 4532	. . 3 |- (T Fn (/) <-> T = (/))
9	7, 8	syl5ibr 224	. 2 |- (T:~H-->~H -> (T = (/) -> ~H = (/)))
10	3, 9	mtoi 122	1 |- (T:~H-->~H -> -. T = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   e. wcel 1300  (/)c0 2875   Fn wfn 3993  -->wf 3994  ~Hchil 10420  0hc0v 10423

This theorem is referenced by:  hmdmadj 11501

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-nul 3445  ax-pow 3481  ax-pr 3524  ax-hv0cl 10505

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-fun 4008  df-fn 4009  df-f 4010

Copyright terms: Public domain