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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 16469 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 4, 6 | sstrd 3742 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | 1, 2, 3, 6 | ismri2d 16466 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 5, 8 | mpbid 222 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
10 | 4 | sseld 3731 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑆)) |
11 | 4 | ssdifd 3877 | . . . . . . 7 ⊢ (𝜑 → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
12 | 6 | ssdifssd 3879 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
13 | 3, 1, 11, 12 | mrcssd 16457 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑇 ∖ {𝑥})) ⊆ (𝑁‘(𝑆 ∖ {𝑥}))) |
14 | 13 | ssneld 3734 | . . . . 5 ⊢ (𝜑 → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) → ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥})))) |
15 | 10, 14 | imim12d 81 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) → (𝑥 ∈ 𝑇 → ¬ 𝑥 ∈ (𝑁‘(𝑇 ∖ {𝑥}))))) |
16 | 15 | ralimdv2 3087 | . . 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 16467 | 1 ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ∖ cdif 3700 ⊆ wss 3703 {csn 4309 ‘cfv 6037 Moorecmre 16415 mrClscmrc 16416 mrIndcmri 16417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-int 4616 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-fv 6045 df-mre 16419 df-mrc 16420 df-mri 16421 |
This theorem is referenced by: mreexexlem2d 16478 acsfiindd 17349 |
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