Theorem ismri2d 17691

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	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
ismri2d.4	⊢ (𝜑 → 𝑆 ⊆ 𝑋)

Assertion

Ref	Expression
ismri2d	⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 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

Step	Hyp	Ref	Expression
1		ismri2d.3	. 2 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
2		ismri2d.4	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
3		ismri2.1	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
4		ismri2.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
5	3, 4	ismri2 17690	. 2 ⊢ ((𝐴 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6	1, 2, 5	syl2anc 583	1 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∖ cdif 3973   ⊆ wss 3976  {csn 4648  ‘cfv 6573  Moorecmre 17640  mrClscmrc 17641  mrIndcmri 17642

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-mre 17644  df-mri 17646

This theorem is referenced by:  ismri2dd  17692  ismri2dad  17695  mrieqvd  17696  mrieqv2d  17697  mrissmrid  17699