Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 4206 | . 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ | |
2 | ch0le.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
3 | 2 | ch0lei 29228 | . . 3 ⊢ 0ℋ ⊆ 𝐴 |
4 | ssid 3989 | . . 3 ⊢ 0ℋ ⊆ 0ℋ | |
5 | 3, 4 | ssini 4208 | . 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ) |
6 | 1, 5 | eqssi 3983 | 1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∩ cin 3935 Cℋ cch 28706 0ℋc0h 28712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-hilex 28776 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-ov 7159 df-sh 28984 df-ch 28998 df-ch0 29030 |
This theorem is referenced by: chm0 29268 |
Copyright terms: Public domain | W3C validator |