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Theorem chocnul 31347
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 4513 . . 3 𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2 0ss 4400 . . . 4 ∅ ⊆ ℋ
3 ocel 31300 . . . 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 710 . 2 (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
65eqriv 2734 1 (⊥‘∅) = ℋ
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951  c0 4333  cfv 6561  (class class class)co 7431  0cc0 11155  chba 30938   ·ih csp 30941  cort 30949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oc 31271
This theorem is referenced by: (None)
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