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Theorem chocnul 24863
Description: Orthogonal complement of the empty set. (Contributed by NM, 31-Oct-2000.) (New usage is discouraged.)
Assertion
Ref Expression
chocnul  |-  ( _|_ `  (/) )  =  ~H

Proof of Theorem chocnul
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 3879 . . 3  |-  A. y  e.  (/)  ( x  .ih  y )  =  0
2 0ss 3761 . . . 4  |-  (/)  C_  ~H
3 ocel 24816 . . . 4  |-  ( (/)  C_ 
~H  ->  ( x  e.  ( _|_ `  (/) )  <->  ( x  e.  ~H  /\  A. y  e.  (/)  ( x  .ih  y )  =  0 ) ) )
42, 3ax-mp 5 . . 3  |-  ( x  e.  ( _|_ `  (/) )  <->  ( x  e.  ~H  /\  A. y  e.  (/)  ( x  .ih  y )  =  0 ) )
51, 4mpbiran2 910 . 2  |-  ( x  e.  ( _|_ `  (/) )  <->  x  e.  ~H )
65eqriv 2447 1  |-  ( _|_ `  (/) )  =  ~H
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2793    C_ wss 3423   (/)c0 3732   ` cfv 5513  (class class class)co 6187   0cc0 9380   ~Hchil 24453    .ih csp 24456   _|_cort 24464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-hilex 24533
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-iota 5476  df-fun 5515  df-fv 5521  df-ov 6190  df-oc 24787
This theorem is referenced by: (None)
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