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Theorem hon0 28522
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 27730 . . 3 0 ∈ ℋ
21n0ii 3903 . 2 ¬ ℋ = ∅
3 fn0 5973 . . 3 (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4 ffn 6007 . . . 4 (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5 fndmu 5955 . . . . 5 ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅)
65ex 450 . . . 4 (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅))
74, 6syl 17 . . 3 (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅))
83, 7syl5bir 233 . 2 (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
92, 8mtoi 190 1 (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  c0 3896   Fn wfn 5847  wf 5848  chil 27646  0c0v 27651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-hv0cl 27730
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-fun 5854  df-fn 5855  df-f 5856
This theorem is referenced by:  hmdmadj  28669
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