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Theorem mrieqv2d 17622
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 4472 . . . . . . 7 (𝑠𝑆 → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
213ad2ant3 1132 . . . . . 6 ((𝜑𝑆𝐼𝑠𝑆) → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
3 mrieqvd.1 . . . . . . . . . 10 (𝜑𝐴 ∈ (Moore‘𝑋))
433ad2ant1 1130 . . . . . . . . 9 ((𝜑𝑆𝐼𝑠𝑆) → 𝐴 ∈ (Moore‘𝑋))
54adantr 479 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝐴 ∈ (Moore‘𝑋))
6 mrieqvd.2 . . . . . . . 8 𝑁 = (mrCls‘𝐴)
7 simprr 771 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥𝑠)
8 difsnb 4811 . . . . . . . . . 10 𝑥𝑠 ↔ (𝑠 ∖ {𝑥}) = 𝑠)
97, 8sylib 217 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) = 𝑠)
10 simpl3 1190 . . . . . . . . . . 11 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1110pssssd 4093 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1211ssdifd 4137 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
139, 12eqsstrrd 4016 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠 ⊆ (𝑆 ∖ {𝑥}))
14 mrieqvd.3 . . . . . . . . . 10 𝐼 = (mrInd‘𝐴)
15 simpl2 1189 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝐼)
1614, 5, 15mrissd 17619 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝑋)
1716ssdifssd 4139 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
185, 6, 13, 17mrcssd 17607 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊆ (𝑁‘(𝑆 ∖ {𝑥})))
19 difssd 4129 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑆)
205, 6, 19, 16mrcssd 17607 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊆ (𝑁𝑆))
215, 6, 16mrcssidd 17608 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆 ⊆ (𝑁𝑆))
22 simprl 769 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥𝑆)
2321, 22sseldd 3977 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥 ∈ (𝑁𝑆))
246, 14, 5, 15, 22ismri2dad 17620 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2520, 23, 24ssnelpssd 4108 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
2618, 25sspsstrd 4104 . . . . . 6 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊊ (𝑁𝑆))
272, 26exlimddv 1930 . . . . 5 ((𝜑𝑆𝐼𝑠𝑆) → (𝑁𝑠) ⊊ (𝑁𝑆))
28273expia 1118 . . . 4 ((𝜑𝑆𝐼) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
2928alrimiv 1922 . . 3 ((𝜑𝑆𝐼) → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
3029ex 411 . 2 (𝜑 → (𝑆𝐼 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
313adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
3231elfvexd 6935 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑋 ∈ V)
33 mrieqvd.4 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3433adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝑋)
3532, 34ssexd 5325 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝑆 ∈ V)
3635difexd 5332 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑆 ∖ {𝑥}) ∈ V)
37 simp1r 1195 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑥𝑆)
38 difsnpss 4812 . . . . . . . . . . . . . . . 16 (𝑥𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆)
3937, 38sylib 217 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑆 ∖ {𝑥}) ⊊ 𝑆)
40 simp2 1134 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠 = (𝑆 ∖ {𝑥}))
4140psseq1d 4088 . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . 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 6900 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) = (𝑁‘(𝑆 ∖ {𝑥})))
4645psseq1d 4088 . . . . . . . . . . . . 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 1118 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥})) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4936, 48spcimdv 3577 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
50493impia 1114 . . . . . . . . 9 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
5150pssned 4094 . . . . . . . 8 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
52513com23 1123 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
5333ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
54333ad2ant1 1130 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑆𝑋)
55 simp3 1135 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
5653, 6, 54, 55mrieqvlemd 17612 . . . . . . . 8 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁𝑆)))
5756necon3bbid 2967 . . . . . . 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 1118 . . . . 5 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑥𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6059ralrimiv 3134 . . . 4 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
6160ex 411 . . 3 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
626, 14, 3, 33ismri2d 17616 . . 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 394  w3a 1084  wal 1531   = wceq 1533  wex 1773  wcel 2098  wne 2929  wral 3050  Vcvv 3461  cdif 3941  wss 3944  wpss 3945  {csn 4630  cfv 6549  Moorecmre 17565  mrClscmrc 17566  mrIndcmri 17567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-mre 17569  df-mrc 17570  df-mri 17571
This theorem is referenced by:  mrissmrcd  17623
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