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| Mirrors > Home > HSE Home > Th. List > chnlen0 | Structured version Visualization version GIF version | ||
| Description: A Hilbert lattice element that is not a subset of another is nonzero. (Contributed by NM, 30-Jun-2004.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| chnlen0 | ⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ch0le 31460 | . . 3 ⊢ (𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵) | |
| 2 | sseq1 4009 | . . 3 ⊢ (𝐴 = 0ℋ → (𝐴 ⊆ 𝐵 ↔ 0ℋ ⊆ 𝐵)) | |
| 3 | 1, 2 | syl5ibrcom 247 | . 2 ⊢ (𝐵 ∈ Cℋ → (𝐴 = 0ℋ → 𝐴 ⊆ 𝐵)) |
| 4 | 3 | con3d 152 | 1 ⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 Cℋ cch 30948 0ℋc0h 30954 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-hilex 31018 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 df-sh 31226 df-ch 31240 df-ch0 31272 |
| This theorem is referenced by: (None) |
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