![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvpss | Structured version Visualization version GIF version |
Description: The covers relation implies proper subset. (cvpss 32314 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 39003 | . . 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 1537 ∈ wcel 2106 ∃wrex 3068 ⊊ wpss 3964 class class class wbr 5148 ‘cfv 6563 LSubSpclss 20947 ⋖L clcv 39000 |
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-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
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-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 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-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lcv 39001 |
This theorem is referenced by: lcvntr 39008 lcvat 39012 lsatcveq0 39014 lsat0cv 39015 lcvexchlem4 39019 lcvexchlem5 39020 lcv1 39023 lsatexch 39025 lsatcvat2 39033 islshpcv 39035 |
Copyright terms: Public domain | W3C validator |