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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 16902 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 4, 6 | sstrd 3970 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | 1, 2, 3, 6 | ismri2d 16899 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 5, 8 | mpbid 234 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
10 | 4 | sseld 3959 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑆)) |
11 | 4 | ssdifd 4110 | . . . . . . 7 ⊢ (𝜑 → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
12 | 6 | ssdifssd 4112 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
13 | 3, 1, 11, 12 | mrcssd 16890 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑇 ∖ {𝑥})) ⊆ (𝑁‘(𝑆 ∖ {𝑥}))) |
14 | 13 | ssneld 3962 | . . . . 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 3175 | . . 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 16900 | 1 ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∖ cdif 3926 ⊆ wss 3929 {csn 4560 ‘cfv 6348 Moorecmre 16848 mrClscmrc 16849 mrIndcmri 16850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-mre 16852 df-mrc 16853 df-mri 16854 |
This theorem is referenced by: mreexexlem2d 16911 acsfiindd 17782 |
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