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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 28300 | . . 3 ⊢ (𝐵 ∈ Cℋ → 0ℋ ⊆ 𝐵) | |
2 | sseq1 3626 | . . 3 ⊢ (𝐴 = 0ℋ → (𝐴 ⊆ 𝐵 ↔ 0ℋ ⊆ 𝐵)) | |
3 | 1, 2 | syl5ibrcom 237 | . 2 ⊢ (𝐵 ∈ Cℋ → (𝐴 = 0ℋ → 𝐴 ⊆ 𝐵)) |
4 | 3 | con3d 148 | 1 ⊢ (𝐵 ∈ Cℋ → (¬ 𝐴 ⊆ 𝐵 → ¬ 𝐴 = 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 Cℋ cch 27786 0ℋc0h 27792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-sh 28064 df-ch 28078 df-ch0 28110 |
This theorem is referenced by: (None) |
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