MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismri2d Structured version   Visualization version   GIF version

Theorem ismri2d 16903
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.)
Hypotheses
Ref Expression
ismri2.1 𝑁 = (mrCls‘𝐴)
ismri2.2 𝐼 = (mrInd‘𝐴)
ismri2d.3 (𝜑𝐴 ∈ (Moore‘𝑋))
ismri2d.4 (𝜑𝑆𝑋)
Assertion
Ref Expression
ismri2d (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆
Allowed substitution hints:   𝜑(𝑥)   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2d
StepHypRef Expression
1 ismri2d.3 . 2 (𝜑𝐴 ∈ (Moore‘𝑋))
2 ismri2d.4 . 2 (𝜑𝑆𝑋)
3 ismri2.1 . . 3 𝑁 = (mrCls‘𝐴)
4 ismri2.2 . . 3 𝐼 = (mrInd‘𝐴)
53, 4ismri2 16902 . 2 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
61, 2, 5syl2anc 586 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208   = wceq 1533  wcel 2110  wral 3138  cdif 3932  wss 3935  {csn 4566  cfv 6354  Moorecmre 16852  mrClscmrc 16853  mrIndcmri 16854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-mre 16856  df-mri 16858
This theorem is referenced by:  ismri2dd  16904  ismri2dad  16907  mrieqvd  16908  mrieqv2d  16909  mrissmrid  16911
  Copyright terms: Public domain W3C validator