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Theorem mrieqv2d 16220
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 4011 . . . . . . 7 (𝑠𝑆 → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
213ad2ant3 1082 . . . . . 6 ((𝜑𝑆𝐼𝑠𝑆) → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
3 mrieqvd.1 . . . . . . . . . 10 (𝜑𝐴 ∈ (Moore‘𝑋))
433ad2ant1 1080 . . . . . . . . 9 ((𝜑𝑆𝐼𝑠𝑆) → 𝐴 ∈ (Moore‘𝑋))
54adantr 481 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝐴 ∈ (Moore‘𝑋))
6 mrieqvd.2 . . . . . . . 8 𝑁 = (mrCls‘𝐴)
7 simprr 795 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥𝑠)
8 difsnb 4306 . . . . . . . . . 10 𝑥𝑠 ↔ (𝑠 ∖ {𝑥}) = 𝑠)
97, 8sylib 208 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) = 𝑠)
10 simpl3 1064 . . . . . . . . . . 11 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1110pssssd 3682 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1211ssdifd 3724 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
139, 12eqsstr3d 3619 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠 ⊆ (𝑆 ∖ {𝑥}))
14 mrieqvd.3 . . . . . . . . . 10 𝐼 = (mrInd‘𝐴)
15 simpl2 1063 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝐼)
1614, 5, 15mrissd 16217 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝑋)
1716ssdifssd 3726 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
185, 6, 13, 17mrcssd 16205 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊆ (𝑁‘(𝑆 ∖ {𝑥})))
19 difssd 3716 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑆)
205, 6, 19, 16mrcssd 16205 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊆ (𝑁𝑆))
215, 6, 16mrcssidd 16206 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆 ⊆ (𝑁𝑆))
22 simprl 793 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥𝑆)
2321, 22sseldd 3584 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥 ∈ (𝑁𝑆))
246, 14, 5, 15, 22ismri2dad 16218 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2520, 23, 24ssnelpssd 3697 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
2618, 25sspsstrd 3693 . . . . . 6 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊊ (𝑁𝑆))
272, 26exlimddv 1860 . . . . 5 ((𝜑𝑆𝐼𝑠𝑆) → (𝑁𝑠) ⊊ (𝑁𝑆))
28273expia 1264 . . . 4 ((𝜑𝑆𝐼) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
2928alrimiv 1852 . . 3 ((𝜑𝑆𝐼) → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
3029ex 450 . 2 (𝜑 → (𝑆𝐼 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
313adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
3231elfvexd 6179 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑋 ∈ V)
33 mrieqvd.4 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3433adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝑋)
3532, 34ssexd 4765 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝑆 ∈ V)
36 difexg 4768 . . . . . . . . . . . 12 (𝑆 ∈ V → (𝑆 ∖ {𝑥}) ∈ V)
3735, 36syl 17 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑆 ∖ {𝑥}) ∈ V)
38 simp1r 1084 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑥𝑆)
39 difsnpss 4307 . . . . . . . . . . . . . . . 16 (𝑥𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆)
4038, 39sylib 208 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑆 ∖ {𝑥}) ⊊ 𝑆)
41 simp2 1060 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠 = (𝑆 ∖ {𝑥}))
4241psseq1d 3677 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆))
4340, 42mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠𝑆)
44 simp3 1061 . . . . . . . . . . . . . 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 6152 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) = (𝑁‘(𝑆 ∖ {𝑥})))
4746psseq1d 3677 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4845, 47mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
49483expia 1264 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥})) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
5037, 49spcimdv 3276 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
51503impia 1258 . . . . . . . . 9 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
5251pssned 3683 . . . . . . . 8 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
53523com23 1268 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
5433ad2ant1 1080 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
55333ad2ant1 1080 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑆𝑋)
56 simp3 1061 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
5754, 6, 55, 56mrieqvlemd 16210 . . . . . . . 8 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁𝑆)))
5857necon3bbid 2827 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))
5953, 58mpbird 247 . . . . . 6 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
60593expia 1264 . . . . 5 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑥𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6160ralrimiv 2959 . . . 4 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
6261ex 450 . . 3 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
636, 14, 3, 33ismri2d 16214 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6462, 63sylibrd 249 . 2 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → 𝑆𝐼))
6530, 64impbid 202 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cdif 3552  wss 3555  wpss 3556  {csn 4148  cfv 5847  Moorecmre 16163  mrClscmrc 16164  mrIndcmri 16165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168  df-mri 16169
This theorem is referenced by:  mrissmrcd  16221
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