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Mirrors > Home > HSE Home > Th. List > chle0 | Structured version Visualization version GIF version |
Description: No Hilbert lattice element is smaller than zero. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
chle0 | ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsh 28774 | . 2 ⊢ (𝐴 ∈ Cℋ → 𝐴 ∈ Sℋ ) | |
2 | shle0 28994 | . 2 ⊢ (𝐴 ∈ Sℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ Cℋ → (𝐴 ⊆ 0ℋ ↔ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 ⊆ wss 3823 Sℋ csh 28478 Cℋ cch 28479 0ℋc0h 28485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 ax-sep 5054 ax-hilex 28549 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rex 3088 df-rab 3091 df-v 3411 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-xp 5407 df-cnv 5409 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fv 6190 df-ov 6973 df-sh 28757 df-ch 28771 df-ch0 28803 |
This theorem is referenced by: chle0i 29004 chssoc 29048 hatomistici 29914 atcvat4i 29949 |
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