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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 29989 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 36037 | . . 3 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) |
8 | 1, 7 | mpbid 233 | . 2 ⊢ (𝜑 → (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))) |
9 | 8 | simpld 495 | 1 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 ⊊ wpss 3934 class class class wbr 5057 ‘cfv 6348 LSubSpclss 19632 ⋖L clcv 36034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-lcv 36035 |
This theorem is referenced by: lcvntr 36042 lcvat 36046 lsatcveq0 36048 lsat0cv 36049 lcvexchlem4 36053 lcvexchlem5 36054 lcv1 36057 lsatexch 36059 lsatcvat2 36067 islshpcv 36069 |
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