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Mirrors > Home > HSE Home > Th. List > ch0lei | Structured version Visualization version GIF version |
Description: The closed subspace zero is the smallest member of Cℋ. (Contributed by NM, 15-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
ch0lei | ⊢ 0ℋ ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ch0le.1 | . 2 ⊢ 𝐴 ∈ Cℋ | |
2 | ch0le 31469 | . 2 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ 0ℋ ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ⊆ wss 3962 Cℋ cch 30957 0ℋc0h 30963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-hilex 31027 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fv 6570 df-ov 7433 df-sh 31235 df-ch 31249 df-ch0 31281 |
This theorem is referenced by: chj0i 31483 chm0i 31518 hst0 32261 |
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