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Mirrors > Home > HSE Home > Th. List > hon0 | Structured version Visualization version GIF version |
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 31032 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | n0ii 4349 | . 2 ⊢ ¬ ℋ = ∅ |
3 | fn0 6700 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
4 | ffn 6737 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
5 | fndmu 6676 | . . . . 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 1537 ∅c0 4339 Fn wfn 6558 ⟶wf 6559 ℋchba 30948 0ℎc0v 30953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-hv0cl 31032 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: hmdmadj 31969 |
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