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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 2835 | . . 3 ⊢ (0ℋ ≠ 𝐴 ↔ 𝐴 ≠ 0ℋ) | |
2 | ch0le 27684 | . . . 4 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) | |
3 | 2 | biantrurd 528 | . . 3 ⊢ (𝐴 ∈ Cℋ → (0ℋ ≠ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
4 | 1, 3 | syl5bbr 273 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
5 | df-pss 3556 | . 2 ⊢ (0ℋ ⊊ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴)) | |
6 | 4, 5 | syl6rbbr 278 | 1 ⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ⊊ wpss 3541 Cℋ cch 27170 0ℋc0h 27176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-hilex 27240 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-ov 6552 df-sh 27448 df-ch 27462 df-ch0 27494 |
This theorem is referenced by: elat2 28583 |
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