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| Description: A Hilbert space operator is not empty. (Contributed by NM, 24-Mar-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hon0 | ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-hv0cl 31022 | . . 3 ⊢ 0ℎ ∈ ℋ | |
| 2 | 1 | n0ii 4343 | . 2 ⊢ ¬ ℋ = ∅ | 
| 3 | fn0 6699 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
| 4 | ffn 6736 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
| 5 | fndmu 6675 | . . . . 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 = ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∅c0 4333 Fn wfn 6556 ⟶wf 6557 ℋchba 30938 0ℎc0v 30943 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hv0cl 31022 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 df-f 6565 | 
| This theorem is referenced by: hmdmadj 31959 | 
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