Theorem hon0 31540

Description: A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
hon0	⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Proof of Theorem hon0

Step	Hyp	Ref	Expression
1		ax-hv0cl 30750	. . 3 ⊢ 0ℎ ∈ ℋ
2	1	n0ii 4329	. 2 ⊢ ¬ ℋ = ∅
3		fn0 6672	. . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4		ffn 6708	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5		fndmu 6647	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅)
6	5	ex 412	. . . 4 ⊢ (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅))
7	4, 6	syl 17	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅))
8	3, 7	biimtrrid 242	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
9	2, 8	mtoi 198	1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533  ∅c0 4315   Fn wfn 6529  ⟶wf 6530   ℋchba 30666  0_ℎc0v 30671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-hv0cl 30750

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-fun 6536  df-fn 6537  df-f 6538

This theorem is referenced by:  hmdmadj  31687