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Theorem hon0 23286
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
StepHypRef Expression
1 ax-hv0cl 22496 . . 3  |-  0h  e.  ~H
2 n0i 3625 . . 3  |-  ( 0h  e.  ~H  ->  -.  ~H  =  (/) )
31, 2ax-mp 8 . 2  |-  -.  ~H  =  (/)
4 fn0 5556 . . 3  |-  ( T  Fn  (/)  <->  T  =  (/) )
5 ffn 5583 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
6 fndmu 5538 . . . . 5  |-  ( ( T  Fn  ~H  /\  T  Fn  (/) )  ->  ~H  =  (/) )
76ex 424 . . . 4  |-  ( T  Fn  ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
85, 7syl 16 . . 3  |-  ( T : ~H --> ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
94, 8syl5bir 210 . 2  |-  ( T : ~H --> ~H  ->  ( T  =  (/)  ->  ~H  =  (/) ) )
103, 9mtoi 171 1  |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725   (/)c0 3620    Fn wfn 5441   -->wf 5442   ~Hchil 22412   0hc0v 22417
This theorem is referenced by:  hmdmadj  23433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hv0cl 22496
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449  df-f 5450
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