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Theorem hon0 23146
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 22356 . . 3  |-  0h  e.  ~H
2 n0i 3578 . . 3  |-  ( 0h  e.  ~H  ->  -.  ~H  =  (/) )
31, 2ax-mp 8 . 2  |-  -.  ~H  =  (/)
4 fn0 5506 . . 3  |-  ( T  Fn  (/)  <->  T  =  (/) )
5 ffn 5533 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
6 fndmu 5488 . . . . 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 1649    e. wcel 1717   (/)c0 3573    Fn wfn 5391   -->wf 5392   ~Hchil 22272   0hc0v 22277
This theorem is referenced by:  hmdmadj  23293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-hv0cl 22356
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-fun 5398  df-fn 5399  df-f 5400
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