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| Mirrors > Home > HSE Home > Th. List > shne0i | Structured version Visualization version GIF version | ||
| Description: A nonzero subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| shne0.1 | ⊢ 𝐴 ∈ Sℋ |
| Ref | Expression |
|---|---|
| shne0i | ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ne 2941 | . 2 ⊢ (𝐴 ≠ 0ℋ ↔ ¬ 𝐴 = 0ℋ) | |
| 2 | df-rex 3071 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
| 3 | nss 4048 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
| 4 | shne0.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
| 5 | shle0 31461 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) | |
| 6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ) |
| 7 | 6 | notbii 320 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ¬ 𝐴 = 0ℋ) |
| 8 | 2, 3, 7 | 3bitr2ri 300 | . 2 ⊢ (¬ 𝐴 = 0ℋ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ) |
| 9 | elch0 31273 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
| 10 | 9 | necon3bbii 2988 | . . 3 ⊢ (¬ 𝑥 ∈ 0ℋ ↔ 𝑥 ≠ 0ℎ) |
| 11 | 10 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) |
| 12 | 1, 8, 11 | 3bitri 297 | 1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3951 0ℎc0v 30943 Sℋ csh 30947 0ℋc0h 30954 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-hilex 31018 ax-hv0cl 31022 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-sh 31226 df-ch0 31272 |
| This theorem is referenced by: chne0i 31472 shatomici 32377 |
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