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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 2938 | . 2 ⊢ (𝐴 ≠ 0ℋ ↔ ¬ 𝐴 = 0ℋ) | |
2 | df-rex 3068 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
3 | nss 4044 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
4 | shne0.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
5 | shle0 31265 | . . . . 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 31077 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
10 | 9 | necon3bbii 2985 | . . 3 ⊢ (¬ 𝑥 ∈ 0ℋ ↔ 𝑥 ≠ 0ℎ) |
11 | 10 | rexbii 3091 | . 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 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3947 0ℎc0v 30747 Sℋ csh 30751 0ℋc0h 30758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-sep 5299 ax-hilex 30822 ax-hv0cl 30826 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-sh 31030 df-ch0 31076 |
This theorem is referenced by: chne0i 31276 shatomici 32181 |
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