Theorem chocnul 31357

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 4519	. . 3 ⊢ ∀𝑦 ∈ ∅ (𝑥 ·ih 𝑦) = 0
2		0ss 4406	. . . 4 ⊢ ∅ ⊆ ℋ
3		ocel 31310	. . . 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 710	. 2 ⊢ (𝑥 ∈ (⊥‘∅) ↔ 𝑥 ∈ ℋ)
6	5	eqriv 2732	1 ⊢ (⊥‘∅) = ℋ

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ⊆ wss 3963  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  0cc0 11153   ℋchba 30948   ·_ih csp 30951  ⊥cort 30959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oc 31281

This theorem is referenced by: (None)