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Mirrors > Home > HSE Home > Th. List > chm0i | Structured version Visualization version GIF version |
Description: Meet with Hilbert lattice zero. (Contributed by NM, 6-Aug-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0le.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
---|---|
chm0i | ⊢ (𝐴 ∩ 0ℋ) = 0ℋ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss2 4228 | . 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ | |
2 | ch0le.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
3 | 2 | ch0lei 31333 | . . 3 ⊢ 0ℋ ⊆ 𝐴 |
4 | ssid 3999 | . . 3 ⊢ 0ℋ ⊆ 0ℋ | |
5 | 3, 4 | ssini 4230 | . 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ) |
6 | 1, 5 | eqssi 3993 | 1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∩ cin 3943 Cℋ cch 30811 0ℋc0h 30817 |
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 2696 ax-sep 5300 ax-hilex 30881 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5684 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fv 6557 df-ov 7422 df-sh 31089 df-ch 31103 df-ch0 31135 |
This theorem is referenced by: chm0 31373 |
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