Theorem hon0 31756

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 30966	. . 3 ⊢ 0ℎ ∈ ℋ
2	1	n0ii 4296	. 2 ⊢ ¬ ℋ = ∅
3		fn0 6617	. . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4		ffn 6656	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5		fndmu 6593	. . . . 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 243	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
9	2, 8	mtoi 199	1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540  ∅c0 4286   Fn wfn 6481  ⟶wf 6482   ℋchba 30882  0_ℎc0v 30887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-hv0cl 30966

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-fun 6488  df-fn 6489  df-f 6490

This theorem is referenced by:  hmdmadj  31903