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Mirrors > Home > MPE Home > Th. List > ismri2d | Structured version Visualization version GIF version |
Description: Criterion for a subset of the base set in a Moore system to be independent. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
ismri2.1 | ⊢ 𝑁 = (mrCls‘𝐴) |
ismri2.2 | ⊢ 𝐼 = (mrInd‘𝐴) |
ismri2d.3 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
ismri2d.4 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
Ref | Expression |
---|---|
ismri2d | ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri2d.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | ismri2d.4 | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
3 | ismri2.1 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
4 | ismri2.2 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
5 | 3, 4 | ismri2 17645 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
6 | 1, 2, 5 | syl2anc 582 | 1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∖ cdif 3944 ⊆ wss 3947 {csn 4633 ‘cfv 6554 Moorecmre 17595 mrClscmrc 17596 mrIndcmri 17597 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
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-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fv 6562 df-mre 17599 df-mri 17601 |
This theorem is referenced by: ismri2dd 17647 ismri2dad 17650 mrieqvd 17651 mrieqv2d 17652 mrissmrid 17654 |
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