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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 28782 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | n0ii 4304 | . 2 ⊢ ¬ ℋ = ∅ |
3 | fn0 6481 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
4 | ffn 6516 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
5 | fndmu 6460 | . . . . 5 ⊢ ((𝑇 Fn ℋ ∧ 𝑇 Fn ∅) → ℋ = ∅) | |
6 | 5 | ex 415 | . . . 4 ⊢ (𝑇 Fn ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
7 | 4, 6 | syl 17 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 Fn ∅ → ℋ = ∅)) |
8 | 3, 7 | syl5bir 245 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 = ∅ → ℋ = ∅)) |
9 | 2, 8 | mtoi 201 | 1 ⊢ (𝑇: ℋ⟶ ℋ → ¬ 𝑇 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∅c0 4293 Fn wfn 6352 ⟶wf 6353 ℋchba 28698 0ℎc0v 28703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-hv0cl 28782 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: hmdmadj 29719 |
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