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Theorem mrieqv2d 16565
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 4199 . . . . . . 7 (𝑠𝑆 → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
213ad2ant3 1165 . . . . . 6 ((𝜑𝑆𝐼𝑠𝑆) → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
3 mrieqvd.1 . . . . . . . . . 10 (𝜑𝐴 ∈ (Moore‘𝑋))
433ad2ant1 1163 . . . . . . . . 9 ((𝜑𝑆𝐼𝑠𝑆) → 𝐴 ∈ (Moore‘𝑋))
54adantr 472 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝐴 ∈ (Moore‘𝑋))
6 mrieqvd.2 . . . . . . . 8 𝑁 = (mrCls‘𝐴)
7 simprr 789 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥𝑠)
8 difsnb 4491 . . . . . . . . . 10 𝑥𝑠 ↔ (𝑠 ∖ {𝑥}) = 𝑠)
97, 8sylib 209 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) = 𝑠)
10 simpl3 1246 . . . . . . . . . . 11 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1110pssssd 3865 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1211ssdifd 3908 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
139, 12eqsstr3d 3800 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠 ⊆ (𝑆 ∖ {𝑥}))
14 mrieqvd.3 . . . . . . . . . 10 𝐼 = (mrInd‘𝐴)
15 simpl2 1244 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝐼)
1614, 5, 15mrissd 16562 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝑋)
1716ssdifssd 3910 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
185, 6, 13, 17mrcssd 16550 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊆ (𝑁‘(𝑆 ∖ {𝑥})))
19 difssd 3900 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑆)
205, 6, 19, 16mrcssd 16550 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊆ (𝑁𝑆))
215, 6, 16mrcssidd 16551 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆 ⊆ (𝑁𝑆))
22 simprl 787 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥𝑆)
2321, 22sseldd 3762 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥 ∈ (𝑁𝑆))
246, 14, 5, 15, 22ismri2dad 16563 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2520, 23, 24ssnelpssd 3880 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
2618, 25sspsstrd 3876 . . . . . 6 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊊ (𝑁𝑆))
272, 26exlimddv 2030 . . . . 5 ((𝜑𝑆𝐼𝑠𝑆) → (𝑁𝑠) ⊊ (𝑁𝑆))
28273expia 1150 . . . 4 ((𝜑𝑆𝐼) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
2928alrimiv 2022 . . 3 ((𝜑𝑆𝐼) → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
3029ex 401 . 2 (𝜑 → (𝑆𝐼 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
313adantr 472 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
3231elfvexd 6410 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑋 ∈ V)
33 mrieqvd.4 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3433adantr 472 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝑋)
3532, 34ssexd 4966 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝑆 ∈ V)
36 difexg 4969 . . . . . . . . . . . 12 (𝑆 ∈ V → (𝑆 ∖ {𝑥}) ∈ V)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑆 ∖ {𝑥}) ∈ V)
38 simp1r 1255 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑥𝑆)
39 difsnpss 4492 . . . . . . . . . . . . . . . 16 (𝑥𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆)
4038, 39sylib 209 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑆 ∖ {𝑥}) ⊊ 𝑆)
41 simp2 1167 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠 = (𝑆 ∖ {𝑥}))
4241psseq1d 3860 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆))
4340, 42mpbird 248 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠𝑆)
44 simp3 1168 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
4543, 44mpd 15 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) ⊊ (𝑁𝑆))
4641fveq2d 6379 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) = (𝑁‘(𝑆 ∖ {𝑥})))
4746psseq1d 3860 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4845, 47mpbid 223 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
49483expia 1150 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥})) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
5037, 49spcimdv 3442 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
51503impia 1145 . . . . . . . . 9 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
5251pssned 3866 . . . . . . . 8 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
53523com23 1156 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
5433ad2ant1 1163 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
55333ad2ant1 1163 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑆𝑋)
56 simp3 1168 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
5754, 6, 55, 56mrieqvlemd 16555 . . . . . . . 8 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁𝑆)))
5857necon3bbid 2974 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))
5953, 58mpbird 248 . . . . . 6 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
60593expia 1150 . . . . 5 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑥𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6160ralrimiv 3112 . . . 4 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
6261ex 401 . . 3 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
636, 14, 3, 33ismri2d 16559 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6462, 63sylibrd 250 . 2 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → 𝑆𝐼))
6530, 64impbid 203 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cdif 3729  wss 3732  wpss 3733  {csn 4334  cfv 6068  Moorecmre 16508  mrClscmrc 16509  mrIndcmri 16510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-mre 16512  df-mrc 16513  df-mri 16514
This theorem is referenced by:  mrissmrcd  16566
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