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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvpss | Structured version Visualization version GIF version | ||
| Description: The covers relation implies proper subset. (cvpss 32304 analog.) (Contributed by NM, 7-Jan-2015.) |
| Ref | Expression |
|---|---|
| lcvfbr.s | ⊢ 𝑆 = (LSubSp‘𝑊) |
| lcvfbr.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) |
| lcvfbr.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) |
| lcvfbr.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) |
| lcvfbr.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) |
| lcvpss.d | ⊢ (𝜑 → 𝑇𝐶𝑈) |
| Ref | Expression |
|---|---|
| lcvpss | ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcvpss.d | . . 3 ⊢ (𝜑 → 𝑇𝐶𝑈) | |
| 2 | lcvfbr.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 3 | lcvfbr.c | . . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
| 4 | lcvfbr.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
| 5 | lcvfbr.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
| 6 | lcvfbr.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 7 | 2, 3, 4, 5, 6 | lcvbr 39022 | . . 3 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) |
| 8 | 1, 7 | mpbid 232 | . 2 ⊢ (𝜑 → (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))) |
| 9 | 8 | simpld 494 | 1 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊊ wpss 3952 class class class wbr 5143 ‘cfv 6561 LSubSpclss 20929 ⋖L clcv 39019 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lcv 39020 |
| This theorem is referenced by: lcvntr 39027 lcvat 39031 lsatcveq0 39033 lsat0cv 39034 lcvexchlem4 39038 lcvexchlem5 39039 lcv1 39042 lsatexch 39044 lsatcvat2 39052 islshpcv 39054 |
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