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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 28786 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | n0ii 4252 | . 2 ⊢ ¬ ℋ = ∅ |
3 | fn0 6451 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
4 | ffn 6487 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
5 | fndmu 6429 | . . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅) | |
6 | 5 | ex 416 | . . . 4 ⊢ (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
8 | 3, 7 | syl5bir 246 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅)) |
9 | 2, 8 | mtoi 202 | 1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∅c0 4243 Fn wfn 6319 ⟶wf 6320 ℋchba 28702 0ℎc0v 28707 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-hv0cl 28786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: hmdmadj 29723 |
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