Theorem hon0 31596

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 30806	. . 3 ⊢ 0ℎ ∈ ℋ
2	1	n0ii 4332	. 2 ⊢ ¬ ℋ = ∅
3		fn0 6680	. . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅)
4		ffn 6716	. . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ)
5		fndmu 6655	. . . . 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 242	. 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅))
9	2, 8	mtoi 198	1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534  ∅c0 4318   Fn wfn 6537  ⟶wf 6538   ℋchba 30722  0_ℎc0v 30727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-hv0cl 30806

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-fun 6544  df-fn 6545  df-f 6546

This theorem is referenced by:  hmdmadj  31743