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Theorem hon0 22389
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 21599 . . 3  |-  0h  e.  ~H
2 n0i 3473 . . 3  |-  ( 0h  e.  ~H  ->  -.  ~H  =  (/) )
31, 2ax-mp 8 . 2  |-  -.  ~H  =  (/)
4 fn0 5379 . . 3  |-  ( T  Fn  (/)  <->  T  =  (/) )
5 ffn 5405 . . . 4  |-  ( T : ~H --> ~H  ->  T  Fn  ~H )
6 fndmu 5361 . . . . 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 1632    e. wcel 1696   (/)c0 3468    Fn wfn 5266   -->wf 5267   ~Hchil 21515   0hc0v 21520
This theorem is referenced by:  hmdmadj  22536
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-hv0cl 21599
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-fun 5273  df-fn 5274  df-f 5275
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