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Mirrors > Home > HSE Home > Th. List > shle0 | Structured version Visualization version GIF version |
Description: No subspace is smaller than the zero subspace. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
shle0 | ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sh0le 29553 | . . 3 ⊢ (𝐴 ∈ Sℋ → 0ℋ ⊆ 𝐴) | |
2 | 1 | biantrud 535 | . 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴))) |
3 | eqss 3933 | . 2 ⊢ (𝐴 = 0ℋ ↔ (𝐴 ⊆ 0ℋ ∧ 0ℋ ⊆ 𝐴)) | |
4 | 2, 3 | bitr4di 292 | 1 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3883 Sℋ csh 29041 0ℋc0h 29048 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 ax-sep 5209 ax-hilex 29112 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5575 df-cnv 5577 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-sh 29320 df-ch0 29366 |
This theorem is referenced by: chle0 29556 shne0i 29561 shs00i 29563 cdj3lem1 30547 |
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