Theorem ismri2dd 17715

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 17714	. 2 ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ ∀𝑥 ∈ 𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
7	1, 6	mpbird 260	1 ⊢ (𝜑 → 𝑆 ∈ 𝐼)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2146  ∀wral 3082   ∖ cdif 3905   ⊆ wss 3908  {csn 4594  ‘cfv 6543  Moorecmre 17659  mrClscmrc 17660  mrIndcmri 17661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-iota 6499  df-fun 6545  df-fv 6551  df-mre 17663  df-mri 17665

This theorem is used by:  mrissmrid  17722  mreexmrid  17724  acsfiindd  18634