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Theorem mrieqv2d 17013
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 4360 . . . . . . 7 (𝑠𝑆 → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
213ad2ant3 1136 . . . . . 6 ((𝜑𝑆𝐼𝑠𝑆) → ∃𝑥(𝑥𝑆 ∧ ¬ 𝑥𝑠))
3 mrieqvd.1 . . . . . . . . . 10 (𝜑𝐴 ∈ (Moore‘𝑋))
433ad2ant1 1134 . . . . . . . . 9 ((𝜑𝑆𝐼𝑠𝑆) → 𝐴 ∈ (Moore‘𝑋))
54adantr 484 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝐴 ∈ (Moore‘𝑋))
6 mrieqvd.2 . . . . . . . 8 𝑁 = (mrCls‘𝐴)
7 simprr 773 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥𝑠)
8 difsnb 4694 . . . . . . . . . 10 𝑥𝑠 ↔ (𝑠 ∖ {𝑥}) = 𝑠)
97, 8sylib 221 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) = 𝑠)
10 simpl3 1194 . . . . . . . . . . 11 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1110pssssd 3988 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠𝑆)
1211ssdifd 4031 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑠 ∖ {𝑥}) ⊆ (𝑆 ∖ {𝑥}))
139, 12eqsstrrd 3916 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑠 ⊆ (𝑆 ∖ {𝑥}))
14 mrieqvd.3 . . . . . . . . . 10 𝐼 = (mrInd‘𝐴)
15 simpl2 1193 . . . . . . . . . 10 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝐼)
1614, 5, 15mrissd 17010 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆𝑋)
1716ssdifssd 4033 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
185, 6, 13, 17mrcssd 16998 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊆ (𝑁‘(𝑆 ∖ {𝑥})))
19 difssd 4023 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑆 ∖ {𝑥}) ⊆ 𝑆)
205, 6, 19, 16mrcssd 16998 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊆ (𝑁𝑆))
215, 6, 16mrcssidd 16999 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑆 ⊆ (𝑁𝑆))
22 simprl 771 . . . . . . . . 9 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥𝑆)
2321, 22sseldd 3878 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → 𝑥 ∈ (𝑁𝑆))
246, 14, 5, 15, 22ismri2dad 17011 . . . . . . . 8 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
2520, 23, 24ssnelpssd 4003 . . . . . . 7 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
2618, 25sspsstrd 3999 . . . . . 6 (((𝜑𝑆𝐼𝑠𝑆) ∧ (𝑥𝑆 ∧ ¬ 𝑥𝑠)) → (𝑁𝑠) ⊊ (𝑁𝑆))
272, 26exlimddv 1942 . . . . 5 ((𝜑𝑆𝐼𝑠𝑆) → (𝑁𝑠) ⊊ (𝑁𝑆))
28273expia 1122 . . . 4 ((𝜑𝑆𝐼) → (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
2928alrimiv 1934 . . 3 ((𝜑𝑆𝐼) → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)))
3029ex 416 . 2 (𝜑 → (𝑆𝐼 → ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
313adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
3231elfvexd 6708 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑋 ∈ V)
33 mrieqvd.4 . . . . . . . . . . . . . 14 (𝜑𝑆𝑋)
3433adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → 𝑆𝑋)
3532, 34ssexd 5192 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝑆 ∈ V)
3635difexd 5197 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑆 ∖ {𝑥}) ∈ V)
37 simp1r 1199 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑥𝑆)
38 difsnpss 4695 . . . . . . . . . . . . . . . 16 (𝑥𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆)
3937, 38sylib 221 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑆 ∖ {𝑥}) ⊊ 𝑆)
40 simp2 1138 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠 = (𝑆 ∖ {𝑥}))
4140psseq1d 3983 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑠𝑆 ↔ (𝑆 ∖ {𝑥}) ⊊ 𝑆))
4239, 41mpbird 260 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → 𝑠𝑆)
43 simp3 1139 . . . . . . . . . . . . . 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 6678 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁𝑠) = (𝑁‘(𝑆 ∖ {𝑥})))
4645psseq1d 3983 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ((𝑁𝑠) ⊊ (𝑁𝑆) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4744, 46mpbid 235 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥}) ∧ (𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
48473expia 1122 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑠 = (𝑆 ∖ {𝑥})) → ((𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
4936, 48spcimdv 3497 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆)))
50493impia 1118 . . . . . . . . 9 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ⊊ (𝑁𝑆))
5150pssned 3989 . . . . . . . 8 ((𝜑𝑥𝑆 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
52513com23 1127 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆))
5333ad2ant1 1134 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝐴 ∈ (Moore‘𝑋))
54333ad2ant1 1134 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑆𝑋)
55 simp3 1139 . . . . . . . . 9 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → 𝑥𝑆)
5653, 6, 54, 55mrieqvlemd 17003 . . . . . . . 8 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) = (𝑁𝑆)))
5756necon3bbid 2971 . . . . . . 7 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → (¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})) ↔ (𝑁‘(𝑆 ∖ {𝑥})) ≠ (𝑁𝑆)))
5852, 57mpbird 260 . . . . . 6 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) ∧ 𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
59583expia 1122 . . . . 5 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → (𝑥𝑆 → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6059ralrimiv 3095 . . . 4 ((𝜑 ∧ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
6160ex 416 . . 3 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
626, 14, 3, 33ismri2d 17007 . . 3 (𝜑 → (𝑆𝐼 ↔ ∀𝑥𝑆 ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥}))))
6361, 62sylibrd 262 . 2 (𝜑 → (∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆)) → 𝑆𝐼))
6430, 63impbid 215 1 (𝜑 → (𝑆𝐼 ↔ ∀𝑠(𝑠𝑆 → (𝑁𝑠) ⊊ (𝑁𝑆))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088  wal 1540   = wceq 1542  wex 1786  wcel 2114  wne 2934  wral 3053  Vcvv 3398  cdif 3840  wss 3843  wpss 3844  {csn 4516  cfv 6339  Moorecmre 16956  mrClscmrc 16957  mrIndcmri 16958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-mre 16960  df-mrc 16961  df-mri 16962
This theorem is referenced by:  mrissmrcd  17014
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