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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 2933 | . 2 ⊢ (𝐴 ≠ 0ℋ ↔ ¬ 𝐴 = 0ℋ) | |
2 | df-rex 3063 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
3 | nss 4039 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
4 | shne0.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
5 | shle0 31190 | . . . . 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 31002 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
10 | 9 | necon3bbii 2980 | . . 3 ⊢ (¬ 𝑥 ∈ 0ℋ ↔ 𝑥 ≠ 0ℎ) |
11 | 10 | rexbii 3086 | . 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 1533 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 ⊆ wss 3941 0ℎc0v 30672 Sℋ csh 30676 0ℋc0h 30683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-hilex 30747 ax-hv0cl 30751 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-sh 30955 df-ch0 31001 |
This theorem is referenced by: chne0i 31201 shatomici 32106 |
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