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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 17644 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | 4, 6 | sstrd 3989 | . 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
8 | 1, 2, 3, 6 | ismri2d 17641 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
9 | 5, 8 | mpbid 231 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
10 | 4 | sseld 3977 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑆)) |
11 | 4 | ssdifd 4137 | . . . . . . 7 ⊢ (𝜑 → (𝑇 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥})) |
12 | 6 | ssdifssd 4139 | . . . . . . 7 ⊢ (𝜑 → (𝑆 ∖ {𝑥}) ⊆ 𝑋) |
13 | 3, 1, 11, 12 | mrcssd 17632 | . . . . . 6 ⊢ (𝜑 → (𝑁‘(𝑇 ∖ {𝑥})) ⊆ (𝑁‘(𝑆 ∖ {𝑥}))) |
14 | 13 | ssneld 3980 | . . . . 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 3153 | . . 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 17642 | 1 ⊢ (𝜑 → 𝑇 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∖ cdif 3943 ⊆ wss 3946 {csn 4623 ‘cfv 6546 Moorecmre 17590 mrClscmrc 17591 mrIndcmri 17592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-mre 17594 df-mrc 17595 df-mri 17596 |
This theorem is referenced by: mreexexlem2d 17653 acsfiindd 18573 |
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