Theorem chocnul 28315

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.

Step	Hyp	Ref	Expression
1		ral0 4109	. . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2		0ss 4005	. . . 4 ⊢ ∅ ⊆ ℋ
3		ocel 28268	. . . 4 ⊢ (∅ ⊆ ℋ → (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0)))
4	2, 3	ax-mp 5	. . 3 ⊢ (𝑥 ∈ (⊥‘∅) ↔ (𝑥 ∈ ℋ ∧ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0))
5	1, 4	mpbiran2 974	. 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
6	5	eqriv 2648	1 ⊢ (⊥‘∅) = ℋ

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  ∅c0 3948  ‘cfv 5926  (class class class)co 6690  0cc0 9974   ℋchil 27904   ·_ih csp 27907  ⊥cort 27915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hilex 27984

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oc 28237

This theorem is referenced by: (None)