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Theorem hon0 30777
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
StepHypRef Expression
1 ax-hv0cl 29987 . . 3 0 ∈ ℋ
21n0ii 4297 . 2 ¬ ℋ = ∅
3 fn0 6633 . . 3 (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4 ffn 6669 . . . 4 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5 fndmu 6610 . . . . 5 ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅)
65ex 414 . . . 4 (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅))
74, 6syl 17 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅))
83, 7biimtrrid 242 . 2 (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
92, 8mtoi 198 1 (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  c0 4283   Fn wfn 6492  wf 6493  chba 29903  0c0v 29908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-hv0cl 29987
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500  df-f 6501
This theorem is referenced by:  hmdmadj  30924
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