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Mirrors > Home > MPE Home > Th. List > mrissmrid | Structured version Visualization version GIF version |
Description: In a Moore system, subsets of independent sets are independent. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrissmrid.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrissmrid.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrissmrid.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
mrissmrid.4 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
mrissmrid.5 | ⊢ (𝜑 → 𝑇 ⊆ 𝑆) |
Ref | Expression |
---|---|
mrissmrid | ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrissmrid.2 | . 2 ⊢ 𝑁 = (mrCls‘𝐴) | |
2 | mrissmrid.3 | . 2 ⊢ 𝐼 = (mrInd‘𝐴) | |
3 | mrissmrid.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
4 | mrissmrid.5 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
5 | mrissmrid.4 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
6 | 2, 3, 5 | mrissd 16119 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 4, 6 | sstrd 3578 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | 1, 2, 3, 6 | ismri2d 16116 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 5, 8 | mpbid 221 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
10 | 4 | sseld 3567 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑆)) |
11 | 4 | ssdifd 3708 | . . . . . . 7 ⊢ (𝜑 → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
12 | 6 | ssdifssd 3710 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
13 | 3, 1, 11, 12 | mrcssd 16107 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑇 ∖ {𝑥})) ⊆ (𝑁‘(𝑆 ∖ {𝑥}))) |
14 | 13 | ssneld 3570 | . . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥})))) |
15 | 10, 14 | imim12d 79 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) → (𝑥 ∈ 𝑇 → ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥}))))) |
16 | 15 | ralimdv2 2944 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ∀𝑥 ∈ 𝑇 ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥})))) |
17 | 9, 16 | mpd 15 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥}))) |
18 | 1, 2, 3, 7, 17 | ismri2dd 16117 | 1 ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ‘cfv 5804 Moorecmre 16065 mrClscmrc 16066 mrIndcmri 16067 |
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-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 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-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-mre 16069 df-mrc 16070 df-mri 16071 |
This theorem is referenced by: mreexexlem2d 16128 acsfiindd 17000 |
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