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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 30064 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 36159 | . . 3 ⊢ (𝜑 → (𝑇𝐶𝑈 ↔ (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈)))) |
8 | 1, 7 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑇 ⊊ 𝑈 ∧ ¬ ∃𝑠 ∈ 𝑆 (𝑇 ⊊ 𝑠 ∧ 𝑠 ⊊ 𝑈))) |
9 | 8 | simpld 497 | 1 ⊢ (𝜑 → 𝑇 ⊊ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 ⊊ wpss 3939 class class class wbr 5068 ‘cfv 6357 LSubSpclss 19705 ⋖L clcv 36156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-lcv 36157 |
This theorem is referenced by: lcvntr 36164 lcvat 36168 lsatcveq0 36170 lsat0cv 36171 lcvexchlem4 36175 lcvexchlem5 36176 lcv1 36179 lsatexch 36181 lsatcvat2 36189 islshpcv 36191 |
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