Theorem hon0 30056

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 29266	. . 3 ⊢ 0ℎ ∈ ℋ
2	1	n0ii 4267	. 2 ⊢ ¬ ℋ = ∅
3		fn0 6548	. . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4		ffn 6584	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5		fndmu 6524	. . . . 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	syl5bir 242	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
9	2, 8	mtoi 198	1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  ∅c0 4253   Fn wfn 6413  ⟶wf 6414   ℋchba 29182  0_ℎc0v 29187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-hv0cl 29266

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-fun 6420  df-fn 6421  df-f 6422

This theorem is referenced by:  hmdmadj  30203