Theorem chocnul 31419

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 4439	. . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2		0ss 4341	. . . 4 ⊢ ∅ ⊆ ℋ
3		ocel 31372	. . . 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 711	. 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
6	5	eqriv 2734	1 ⊢ (⊥‘∅) = ℋ

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3890  ∅c0 4274  ‘cfv 6490  (class class class)co 7358  0cc0 11027   ℋchba 31010   ·_ih csp 31013  ⊥cort 31021

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368  ax-hilex 31090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-oc 31343

This theorem is referenced by: (None)