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Mirrors > Home > HSE Home > Th. List > ch0pss | Structured version Visualization version GIF version |
Description: The zero subspace is a proper subset of nonzero Hilbert lattice elements. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ch0pss | ⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necom 3066 | . . 3 ⊢ (0ℋ ≠ 𝐴 ↔ 𝐴 ≠ 0ℋ) | |
2 | ch0le 29145 | . . . 4 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) | |
3 | 2 | biantrurd 533 | . . 3 ⊢ (𝐴 ∈ Cℋ → (0ℋ ≠ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
4 | 1, 3 | syl5bbr 286 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
5 | df-pss 3951 | . 2 ⊢ (0ℋ ⊊ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴)) | |
6 | 4, 5 | syl6rbbr 291 | 1 ⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ⊊ wpss 3934 Cℋ cch 28633 0ℋc0h 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-hilex 28703 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fv 6356 df-ov 7148 df-sh 28911 df-ch 28925 df-ch0 28957 |
This theorem is referenced by: elat2 30044 |
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