Theorem ismri2 17683

Description: Criterion for a subset of the base set in a Moore system to be independent. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ismri2.1	⊢ 𝑁 = (mrCls‘𝐴)
ismri2.2	⊢ 𝐼 = (mrInd‘𝐴)

Assertion

Ref	Expression
ismri2	⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆

Allowed substitution hints:   𝐼(𝑥)   𝑁(𝑥)   𝑋(𝑥)

Proof of Theorem ismri2

Step	Hyp	Ref	Expression
1		ismri2.1	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
2		ismri2.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
3	1, 2	ismri 17682	. 2 ⊢ (𝐴 ∈ (Moore‘𝑋) → (𝑆 ∈ 𝐼 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))))
4	3	baibd 539	1 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1538   ∈ wcel 2107  ∀wral 3060   ∖ cdif 3961   ⊆ wss 3964  {csn 4632  ‘cfv 6566  Moorecmre 17633  mrClscmrc 17634  mrIndcmri 17635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pow 5372  ax-pr 5439  ax-un 7758

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-nf 1782  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-in 3971  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5584  df-xp 5696  df-rel 5697  df-cnv 5698  df-co 5699  df-dm 5700  df-rn 5701  df-iota 6519  df-fun 6568  df-fv 6574  df-mre 17637  df-mri 17639

This theorem is referenced by:  ismri2d  17684  lindsdom  37613  aacllem  49053