Theorem ismri2dd 17647

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	⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
ismri2d.4	⊢ (𝜑 → 𝑆 ⊆ 𝑋)
ismri2dd.5	⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))

Assertion

Ref	Expression
ismri2dd	⊢ (𝜑 → 𝑆 ∈ 𝐼)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑆

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

Proof of Theorem ismri2dd

Step	Hyp	Ref	Expression
1		ismri2dd.5	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2		ismri2.1	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
3		ismri2.2	. . 3 ⊢ 𝐼 = (mrInd‘𝐴)
4		ismri2d.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5		ismri2d.4	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
6	2, 3, 4, 5	ismri2d 17646	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbird 259	1 ⊢ (𝜑 → 𝑆 ∈ 𝐼)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ∖ cdif 3901   ⊆ wss 3904  {csn 4581  ‘cfv 6515  Moorecmre 17591  mrClscmrc 17592  mrIndcmri 17593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6471  df-fun 6517  df-fv 6523  df-mre 17595  df-mri 17597

This theorem is referenced by:  mrissmrid  17654  mreexmrid  17656  acsfiindd  18566