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Theorem ismri2dd 16900
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 16899 . 2 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
71, 6mpbird 260 1 (𝜑𝑆𝐼)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1538  wcel 2112  wral 3109  cdif 3881  wss 3884  {csn 4528  cfv 6328  Moorecmre 16848  mrClscmrc 16849  mrIndcmri 16850
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fv 6336  df-mre 16852  df-mri 16854
This theorem is referenced by:  mrissmrid  16907  mreexmrid  16909  acsfiindd  17782
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