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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 4126 | . 2 ⊢ (𝐴 ∩ 0ℋ) ⊆ 0ℋ | |
2 | ch0le.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
3 | 2 | ch0lei 28919 | . . 3 ⊢ 0ℋ ⊆ 𝐴 |
4 | ssid 3910 | . . 3 ⊢ 0ℋ ⊆ 0ℋ | |
5 | 3, 4 | ssini 4128 | . 2 ⊢ 0ℋ ⊆ (𝐴 ∩ 0ℋ) |
6 | 1, 5 | eqssi 3905 | 1 ⊢ (𝐴 ∩ 0ℋ) = 0ℋ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 ∈ wcel 2081 ∩ cin 3858 Cℋ cch 28397 0ℋc0h 28403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 ax-sep 5094 ax-hilex 28467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fv 6233 df-ov 7019 df-sh 28675 df-ch 28689 df-ch0 28721 |
This theorem is referenced by: chm0 28959 |
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