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Mirrors > Home > MPE Home > Th. List > ismri2dd | Structured version Visualization version GIF version |
Description: Definition of independence of a subset of the base set in a Moore system. One-way 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) |
ismri2dd.5 | ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) |
Ref | Expression |
---|---|
ismri2dd | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismri2dd.5 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))) | |
2 | ismri2.1 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | ismri2.2 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | ismri2d.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
5 | ismri2d.4 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 2, 3, 4, 5 | ismri2d 17090 | . 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
7 | 1, 6 | mpbird 260 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 ‘cfv 6358 Moorecmre 17039 mrClscmrc 17040 mrIndcmri 17041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-iota 6316 df-fun 6360 df-fv 6366 df-mre 17043 df-mri 17045 |
This theorem is referenced by: mrissmrid 17098 mreexmrid 17100 acsfiindd 18013 |
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