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Theorem ismri2dd 16609
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.)
Hypotheses
Ref Expression
ismri2.1 𝑁 = (mrCls‘𝐴)
ismri2.2 𝐼 = (mrInd‘𝐴)
ismri2d.3 (𝜑𝐴 ∈ (Moore‘𝑋))
ismri2d.4 (𝜑𝑆𝑋)
ismri2dd.5 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
Assertion
Ref Expression
ismri2dd (𝜑𝑆𝐼)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆
Allowed substitution hints:   𝜑(𝑥)   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2dd
StepHypRef Expression
1 ismri2dd.5 . 2 (𝜑 → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2 ismri2.1 . . 3 𝑁 = (mrCls‘𝐴)
3 ismri2.2 . . 3 𝐼 = (mrInd‘𝐴)
4 ismri2d.3 . . 3 (𝜑𝐴 ∈ (Moore‘𝑋))
5 ismri2d.4 . . 3 (𝜑𝑆𝑋)
62, 3, 4, 5ismri2d 16608 . 2 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbird 249 1 (𝜑𝑆𝐼)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = 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:  mrissmrid  16616  mreexmrid  16618  acsfiindd  17492
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