Theorem chocnul 29739

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 4449	. . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2		0ss 4336	. . . 4 ⊢ ∅ ⊆ ℋ
3		ocel 29692	. . . 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 708	. 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
6	5	eqriv 2733	1 ⊢ (⊥‘∅) = ℋ

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3061   ⊆ wss 3892  ∅c0 4262  ‘cfv 6458  (class class class)co 7307  0cc0 10921   ℋchba 29330   ·_ih csp 29333  ⊥cort 29341

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-hilex 29410

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-iota 6410  df-fun 6460  df-fv 6466  df-ov 7310  df-oc 29663

This theorem is referenced by: (None)