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Theorem chocnul 22787
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 3696 . . 3  |-  A. y  e.  (/)  ( x  .ih  y )  =  0
2 0ss 3620 . . . 4  |-  (/)  C_  ~H
3 ocel 22740 . . . 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 886 . 2  |-  ( x  e.  ( _|_ `  (/) )  <->  x  e.  ~H )
65eqriv 2405 1  |-  ( _|_ `  (/) )  =  ~H
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670    C_ wss 3284   (/)c0 3592   ` cfv 5417  (class class class)co 6044   0cc0 8950   ~Hchil 22379    .ih csp 22382   _|_cort 22390
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-hilex 22459
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-iota 5381  df-fun 5419  df-fv 5425  df-ov 6047  df-oc 22711
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