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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 28528 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 33326 | . . 3 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) |
8 | 1, 7 | mpbid 221 | . 2 ⊢ (𝜑 → (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))) |
9 | 8 | simpld 474 | 1 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊊ wpss 3541 class class class wbr 4583 ‘cfv 5804 LSubSpclss 18753 ⋖L clcv 33323 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-lcv 33324 |
This theorem is referenced by: lcvntr 33331 lcvat 33335 lsatcveq0 33337 lsat0cv 33338 lcvexchlem4 33342 lcvexchlem5 33343 lcv1 33346 lsatexch 33348 lsatcvat2 33356 islshpcv 33358 |
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