Theorem hon0 31033

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 30243	. . 3 ⊢ 0ℎ ∈ ℋ
2	1	n0ii 4335	. 2 ⊢ ¬ ℋ = ∅
3		fn0 6678	. . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4		ffn 6714	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5		fndmu 6653	. . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅)
6	5	ex 413	. . . 4 ⊢ (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅))
7	4, 6	syl 17	. . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅))
8	3, 7	biimtrrid 242	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
9	2, 8	mtoi 198	1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541  ∅c0 4321   Fn wfn 6535  ⟶wf 6536   ℋchba 30159  0_ℎc0v 30164

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-hv0cl 30243

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-fun 6542  df-fn 6543  df-f 6544

This theorem is referenced by:  hmdmadj  31180