Theorem hon0 28652

Description: A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Assertion

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

Proof of Theorem hon0

Step	Hyp	Ref	Expression
1		ax-hv0cl 27860	. . 3 |- 0h e. ~H
2	1	n0ii 3922	. 2 |- -. ~H = (/)
3		fn0 6011	. . 3 |- ( T Fn (/) <-> T = (/) )
4		ffn 6045	. . . 4 |- ( T : ~H --> ~H -> T Fn ~H )
5		fndmu 5992	. . . . 5 |- ( ( T Fn ~H /\ T Fn (/) ) -> ~H = (/) )
6	5	ex 450	. . . 4 |- ( T Fn ~H -> ( T Fn (/) -> ~H = (/) ) )
7	4, 6	syl 17	. . 3 |- ( T : ~H --> ~H -> ( T Fn (/) -> ~H = (/) ) )
8	3, 7	syl5bir 233	. 2 |- ( T : ~H --> ~H -> ( T = (/) -> ~H = (/) ) )
9	2, 8	mtoi 190	1 |- ( T : ~H --> ~H -> -. T = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   (/)c0 3915   Fn wfn 5883   -->wf 5884   ~Hchil 27776   0hc0v 27781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hv0cl 27860

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891  df-f 5892

This theorem is referenced by:  hmdmadj  28799