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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 16907 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 4, 6 | sstrd 3977 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | 1, 2, 3, 6 | ismri2d 16904 | . . . 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 3966 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑆)) |
11 | 4 | ssdifd 4117 | . . . . . . 7 ⊢ (𝜑 → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
12 | 6 | ssdifssd 4119 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
13 | 3, 1, 11, 12 | mrcssd 16895 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑇 ∖ {𝑥})) ⊆ (𝑁‘(𝑆 ∖ {𝑥}))) |
14 | 13 | ssneld 3969 | . . . . 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 3176 | . . 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 16905 | 1 ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Moorecmre 16853 mrClscmrc 16854 mrIndcmri 16855 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-mre 16857 df-mrc 16858 df-mri 16859 |
This theorem is referenced by: mreexexlem2d 16916 acsfiindd 17787 |
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