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Theorem mrieqv2d 17348
Description: In a Moore system, a set is independent if and only if all its proper subsets have closure properly contained in the closure of the set. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mrieqvd.1 (𝜑𝐴 ∈ (Moore‘𝑋))
mrieqvd.2 𝑁 = (mrCls‘𝐴)
mrieqvd.3 𝐼 = (mrInd‘𝐴)
mrieqvd.4 (𝜑𝑆𝑋)
Assertion
Ref Expression
mrieqv2d (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
Distinct variable groups:   𝑆,𝑠   𝜑,𝑠   𝐼,𝑠   𝑁,𝑠
Allowed substitution hints:   𝐴(𝑠)   𝑋(𝑠)

Proof of Theorem mrieqv2d
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pssnel 4404 . . . . . . 7 (𝑠𝑆 → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
213ad2ant3 1134 . . . . . 6 ((𝜑𝑆𝐼𝑠𝑆) → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
3 mrieqvd.1 . . . . . . . . . 10 (𝜑𝐴 ∈ (Moore‘𝑋))
433ad2ant1 1132 . . . . . . . . 9 ((𝜑𝑆𝐼𝑠𝑆) → 𝐴 ∈ (Moore‘𝑋))
54adantr 481 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝐴 ∈ (Moore‘𝑋))
6 mrieqvd.2 . . . . . . . 8 𝑁 = (mrCls‘𝐴)
7 simprr 770 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥𝑠)
8 difsnb 4739 . . . . . . . . . 10 𝑥𝑠 ↔ (𝑠 ∖ {𝑥}) = 𝑠)
97, 8sylib 217 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) = 𝑠)
10 simpl3 1192 . . . . . . . . . . 11 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1110pssssd 4032 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1211ssdifd 4075 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
139, 12eqsstrrd 3960 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠 ⊆ (𝑆 ∖ {𝑥}))
14 mrieqvd.3 . . . . . . . . . 10 𝐼 = (mrInd‘𝐴)
15 simpl2 1191 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝐼)
1614, 5, 15mrissd 17345 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝑋)
1716ssdifssd 4077 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
185, 6, 13, 17mrcssd 17333 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊆ (𝑁‘(𝑆 ∖ {𝑥})))
19 difssd 4067 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑆)
205, 6, 19, 16mrcssd 17333 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊆ (𝑁𝑆))
215, 6, 16mrcssidd 17334 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆 ⊆ (𝑁𝑆))
22 simprl 768 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥𝑆)
2321, 22sseldd 3922 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥 ∈ (𝑁𝑆))
246, 14, 5, 15, 22ismri2dad 17346 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2520, 23, 24ssnelpssd 4047 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
2618, 25sspsstrd 4043 . . . . . 6 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊊ (𝑁𝑆))
272, 26exlimddv 1938 . . . . 5 ((𝜑𝑆𝐼𝑠𝑆) → (𝑁𝑠) ⊊ (𝑁𝑆))
28273expia 1120 . . . 4 ((𝜑𝑆𝐼) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
2928alrimiv 1930 . . 3 ((𝜑𝑆𝐼) → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
3029ex 413 . 2 (𝜑 → (𝑆𝐼 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
313adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
3231elfvexd 6808 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑋 ∈ V)
33 mrieqvd.4 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3433adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝑋)
3532, 34ssexd 5248 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝑆 ∈ V)
3635difexd 5253 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑆 ∖ {𝑥}) ∈ V)
37 simp1r 1197 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑥𝑆)
38 difsnpss 4740 . . . . . . . . . . . . . . . 16 (𝑥𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆)
3937, 38sylib 217 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑆 ∖ {𝑥}) ⊊ 𝑆)
40 simp2 1136 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠 = (𝑆 ∖ {𝑥}))
4140psseq1d 4027 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆))
4239, 41mpbird 256 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠𝑆)
43 simp3 1137 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
4442, 43mpd 15 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) ⊊ (𝑁𝑆))
4540fveq2d 6778 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) = (𝑁‘(𝑆 ∖ {𝑥})))
4645psseq1d 4027 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4744, 46mpbid 231 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
48473expia 1120 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥})) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4936, 48spcimdv 3532 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
50493impia 1116 . . . . . . . . 9 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
5150pssned 4033 . . . . . . . 8 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
52513com23 1125 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
5333ad2ant1 1132 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
54333ad2ant1 1132 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑆𝑋)
55 simp3 1137 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
5653, 6, 54, 55mrieqvlemd 17338 . . . . . . . 8 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁𝑆)))
5756necon3bbid 2981 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))
5852, 57mpbird 256 . . . . . 6 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
59583expia 1120 . . . . 5 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑥𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6059ralrimiv 3102 . . . 4 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
6160ex 413 . . 3 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
626, 14, 3, 33ismri2d 17342 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6361, 62sylibrd 258 . 2 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → 𝑆𝐼))
6430, 63impbid 211 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086  wal 1537   = wceq 1539  wex 1782  wcel 2106  wne 2943  wral 3064  Vcvv 3432  cdif 3884  wss 3887  wpss 3888  {csn 4561  cfv 6433  Moorecmre 17291  mrClscmrc 17292  mrIndcmri 17293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-mre 17295  df-mrc 17296  df-mri 17297
This theorem is referenced by:  mrissmrcd  17349
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