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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 16607 | . 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
6 | 1, 2, 5 | syl2anc 580 | 1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∀wral 3089 ∖ cdif 3766 ⊆ wss 3769 {csn 4368 ‘cfv 6101 Moorecmre 16557 mrClscmrc 16558 mrIndcmri 16559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-iota 6064 df-fun 6103 df-fv 6109 df-mre 16561 df-mri 16563 |
This theorem is referenced by: ismri2dd 16609 ismri2dad 16612 mrieqvd 16613 mrieqv2d 16614 mrissmrid 16616 |
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