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Theorem hon0 22373
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 21583 . . 3  |-  0h  e.  ~H
2 n0i 3460 . . 3  |-  ( 0h  e.  ~H  ->  -.  ~H  =  (/) )
31, 2ax-mp 8 . 2  |-  -.  ~H  =  (/)
4 fn0 5363 . . 3  |-  ( T  Fn  (/)  <->  T  =  (/) )
5 ffn 5389 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
6 fndmu 5345 . . . . 5  |-  ( ( T  Fn  ~H  /\  T  Fn  (/) )  ->  ~H  =  (/) )
76ex 423 . . . 4  |-  ( T  Fn  ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
85, 7syl 15 . . 3  |-  ( T : ~H --> ~H  ->  ( T  Fn  (/)  ->  ~H  =  (/) ) )
94, 8syl5bir 209 . 2  |-  ( T : ~H --> ~H  ->  ( T  =  (/)  ->  ~H  =  (/) ) )
103, 9mtoi 169 1  |-  ( T : ~H --> ~H  ->  -.  T  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   (/)c0 3455    Fn wfn 5250   -->wf 5251   ~Hchil 21499   0hc0v 21504
This theorem is referenced by:  hmdmadj  22520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-hv0cl 21583
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258  df-f 5259
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