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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 | df-pss 3968 | . 2 ⊢ (0ℋ ⊊ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴)) | |
2 | necom 2995 | . . 3 ⊢ (0ℋ ≠ 𝐴 ↔ 𝐴 ≠ 0ℋ) | |
3 | ch0le 30694 | . . . 4 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) | |
4 | 3 | biantrurd 534 | . . 3 ⊢ (𝐴 ∈ Cℋ → (0ℋ ≠ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
5 | 2, 4 | bitr3id 285 | . 2 ⊢ (𝐴 ∈ Cℋ → (𝐴 ≠ 0ℋ ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴))) |
6 | 1, 5 | bitr4id 290 | 1 ⊢ (𝐴 ∈ Cℋ → (0ℋ ⊊ 𝐴 ↔ 𝐴 ≠ 0ℋ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ≠ wne 2941 ⊆ wss 3949 ⊊ wpss 3950 Cℋ cch 30182 0ℋc0h 30188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-hilex 30252 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fv 6552 df-ov 7412 df-sh 30460 df-ch 30474 df-ch0 30506 |
This theorem is referenced by: elat2 31593 |
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