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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 3983 | . 2 ⊢ (0ℋ ⊊ 𝐴 ↔ (0ℋ ⊆ 𝐴 ∧ 0ℋ ≠ 𝐴)) | |
2 | necom 2992 | . . 3 ⊢ (0ℋ ≠ 𝐴 ↔ 𝐴 ≠ 0ℋ) | |
3 | ch0le 31470 | . . . 4 ⊢ (𝐴 ∈ Cℋ → 0ℋ ⊆ 𝐴) | |
4 | 3 | biantrurd 532 | . . 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 206 ∧ wa 395 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ⊊ wpss 3964 Cℋ cch 30958 0ℋc0h 30964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-hilex 31028 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fv 6571 df-ov 7434 df-sh 31236 df-ch 31250 df-ch0 31282 |
This theorem is referenced by: elat2 32369 |
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