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Theorem chocnul 22021
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 3634 . . 3  |-  A. y  e.  (/)  ( x  .ih  y )  =  0
2 0ss 3559 . . . 4  |-  (/)  C_  ~H
3 ocel 21974 . . . 4  |-  ( (/)  C_ 
~H  ->  ( x  e.  ( _|_ `  (/) )  <->  ( x  e.  ~H  /\  A. y  e.  (/)  ( x  .ih  y )  =  0 ) ) )
42, 3ax-mp 8 . . 3  |-  ( x  e.  ( _|_ `  (/) )  <->  ( x  e.  ~H  /\  A. y  e.  (/)  ( x  .ih  y )  =  0 ) )
51, 4mpbiran2 885 . 2  |-  ( x  e.  ( _|_ `  (/) )  <->  x  e.  ~H )
65eqriv 2355 1  |-  ( _|_ `  (/) )  =  ~H
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619    C_ wss 3228   (/)c0 3531   ` cfv 5337  (class class class)co 5945   0cc0 8827   ~Hchil 21613    .ih csp 21616   _|_cort 21624
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-hilex 21693
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oc 21945
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