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Theorem chocnul 27571
 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 4028 . . 3 𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2 0ss 3924 . . . 4 ∅ ⊆ ℋ
3 ocel 27524 . . . 4 (∅ ⊆ ℋ → (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0)))
42, 3ax-mp 5 . . 3 (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0))
51, 4mpbiran2 956 . 2 (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
65eqriv 2607 1 (⊥‘∅) = ℋ
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ℋchil 27160   ·ih csp 27163  ⊥cort 27171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oc 27493 This theorem is referenced by: (None)
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