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Theorem ismri2d 16209
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 16208 . 2 ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
61, 2, 5syl2anc 692 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196   = wceq 1480  wcel 1992  wral 2912  cdif 3557  wss 3560  {csn 4153  cfv 5850  Moorecmre 16158  mrClscmrc 16159  mrIndcmri 16160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858  df-mre 16162  df-mri 16164
This theorem is referenced by:  ismri2dd  16210  ismri2dad  16213  mrieqvd  16214  mrieqv2d  16215  mrissmrid  16217
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