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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 28774 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | 1 | n0ii 4301 | . 2 ⊢ ¬ ℋ = ∅ |
3 | fn0 6473 | . . 3 ⊢ (𝑇 Fn ∅ ↔ 𝑇 = ∅) | |
4 | ffn 6508 | . . . 4 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
5 | fndmu 6452 | . . . . 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 1533 ∅c0 4290 Fn wfn 6344 ⟶wf 6345 ℋchba 28690 0ℎc0v 28695 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-hv0cl 28774 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-fun 6351 df-fn 6352 df-f 6353 |
This theorem is referenced by: hmdmadj 29711 |
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