Theorem hon0 9719

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 8873	. . 3 |- 0h e. H~
2		n0i 2285	. . 3 |- (0h e. H~ -> -. H~ = (/))
3	1, 2	ax-mp 7	. 2 |- -. H~ = (/)
4		ffn 3627	. . . 4 |- (T:H~-->H~ -> T Fn H~)
5		fndmu 3589	. . . . 5 |- ((T Fn H~ /\ T Fn (/)) -> H~ = (/))
6	5	ex 373	. . . 4 |- (T Fn H~ -> (T Fn (/) -> H~ = (/)))
7	4, 6	syl 10	. . 3 |- (T:H~-->H~ -> (T Fn (/) -> H~ = (/)))
8		fn0 3605	. . 3 |- (T Fn (/) <-> T = (/))
9	7, 8	syl5ibr 207	. 2 |- (T:H~-->H~ -> (T = (/) -> H~ = (/)))
10	3, 9	mtoi 107	1 |- (T:H~-->H~ -> -. T = (/))

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958  (/)c0 2280   Fn wfn 3177  -->wf 3178  H~chil 8788  0hc0v 8791

This theorem is referenced by:  hmdmadjt 9864

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-hv0cl 8873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-fun 3192  df-fn 3193  df-f 3194

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