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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 17510 | . 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))) |
7 | 1, 6 | mpbird 256 | 1 ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1541 ∈ wcel 2106 ∀wral 3063 ∖ cdif 3906 ⊆ wss 3909 {csn 4585 ‘cfv 6494 Moorecmre 17459 mrClscmrc 17460 mrIndcmri 17461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fv 6502 df-mre 17463 df-mri 17465 |
This theorem is referenced by: mrissmrid 17518 mreexmrid 17520 acsfiindd 18439 |
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