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Theorem ismri 17579
Description: Criterion for a set to be an independent set of a Moore system. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ismri.1 𝑁 = (mrClsβ€˜π΄)
ismri.2 𝐼 = (mrIndβ€˜π΄)
Assertion
Ref Expression
ismri (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝐼 ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑆
Allowed substitution hints:   𝐼(π‘₯)   𝑁(π‘₯)   𝑋(π‘₯)

Proof of Theorem ismri
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 ismri.1 . . . . 5 𝑁 = (mrClsβ€˜π΄)
2 ismri.2 . . . . 5 𝐼 = (mrIndβ€˜π΄)
31, 2mrisval 17578 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝐼 = {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))})
43eleq2d 2817 . . 3 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝐼 ↔ 𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))}))
5 difeq1 4114 . . . . . . . 8 (𝑠 = 𝑆 β†’ (𝑠 βˆ– {π‘₯}) = (𝑆 βˆ– {π‘₯}))
65fveq2d 6894 . . . . . . 7 (𝑠 = 𝑆 β†’ (π‘β€˜(𝑠 βˆ– {π‘₯})) = (π‘β€˜(𝑆 βˆ– {π‘₯})))
76eleq2d 2817 . . . . . 6 (𝑠 = 𝑆 β†’ (π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯})) ↔ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
87notbid 317 . . . . 5 (𝑠 = 𝑆 β†’ (Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯})) ↔ Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
98raleqbi1dv 3331 . . . 4 (𝑠 = 𝑆 β†’ (βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯})) ↔ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
109elrab 3682 . . 3 (𝑆 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ βˆ€π‘₯ ∈ 𝑠 Β¬ π‘₯ ∈ (π‘β€˜(𝑠 βˆ– {π‘₯}))} ↔ (𝑆 ∈ 𝒫 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))))
114, 10bitrdi 286 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ 𝒫 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))))
12 elfvex 6928 . . . 4 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ 𝑋 ∈ V)
13 elpw2g 5343 . . . 4 (𝑋 ∈ V β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1412, 13syl 17 . . 3 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1514anbi1d 628 . 2 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ ((𝑆 ∈ 𝒫 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯}))) ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))))
1611, 15bitrd 278 1 (𝐴 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 ∈ 𝐼 ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑆 Β¬ π‘₯ ∈ (π‘β€˜(𝑆 βˆ– {π‘₯})))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  π’« cpw 4601  {csn 4627  β€˜cfv 6542  Moorecmre 17530  mrClscmrc 17531  mrIndcmri 17532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-mre 17534  df-mri 17536
This theorem is referenced by:  ismri2  17580  mriss  17583  lbsacsbs  20914
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