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Theorem hon0 29574
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 28784 . . 3 0 ∈ ℋ
21n0ii 4274 . 2 ¬ ℋ = ∅
3 fn0 6459 . . 3 (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4 ffn 6494 . . . 4 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5 fndmu 6438 . . . . 5 ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅)
65ex 416 . . . 4 (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅))
74, 6syl 17 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅))
83, 7syl5bir 246 . 2 (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
92, 8mtoi 202 1 (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  c0 4265   Fn wfn 6329  wf 6330  chba 28700  0c0v 28705
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-hv0cl 28784
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-fun 6336  df-fn 6337  df-f 6338
This theorem is referenced by:  hmdmadj  29721
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