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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 28436 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | n0ii 4151 | . 2 ⊢ ¬ ℋ = ∅ |
3 | fn0 6259 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
4 | ffn 6293 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
5 | fndmu 6240 | . . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅) | |
6 | 5 | ex 403 | . . . 4 ⊢ (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
8 | 3, 7 | syl5bir 235 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅)) |
9 | 2, 8 | mtoi 191 | 1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1601 ∅c0 4141 Fn wfn 6132 ⟶wf 6133 ℋchba 28352 0ℎc0v 28357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-hv0cl 28436 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-fun 6139 df-fn 6140 df-f 6141 |
This theorem is referenced by: hmdmadj 29375 |
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