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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 2988 | . 2 ⊢ (𝐴 ≠ 0ℋ ↔ ¬ 𝐴 = 0ℋ) | |
2 | df-rex 3112 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
3 | nss 3977 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ 0ℋ)) | |
4 | shne0.1 | . . . . 5 ⊢ 𝐴 ∈ Sℋ | |
5 | shle0 29225 | . . . . 5 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) | |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ) |
7 | 6 | notbii 323 | . . 3 ⊢ (¬ 𝐴 ⊆ 0ℋ ↔ ¬ 𝐴 = 0ℋ) |
8 | 2, 3, 7 | 3bitr2ri 303 | . 2 ⊢ (¬ 𝐴 = 0ℋ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ) |
9 | elch0 29037 | . . . 4 ⊢ (𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ) | |
10 | 9 | necon3bbii 3034 | . . 3 ⊢ (¬ 𝑥 ∈ 0ℋ ↔ 𝑥 ≠ 0ℎ) |
11 | 10 | rexbii 3210 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) |
12 | 1, 8, 11 | 3bitri 300 | 1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 ⊆ wss 3881 0ℎc0v 28707 Sℋ csh 28711 0ℋc0h 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-hilex 28782 ax-hv0cl 28786 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sh 28990 df-ch0 29036 |
This theorem is referenced by: chne0i 29236 shatomici 30141 |
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