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Theorem mreexmrid 16505
Description: In a Moore system whose closure operator has the exchange property, if a set is independent and an element is not in its closure, then adding the element to the set gives another independent set. Lemma 4.1.5 in [FaureFrolicher] p. 84. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
mreexmrid.1 (𝜑𝐴 ∈ (Moore‘𝑋))
mreexmrid.2 𝑁 = (mrCls‘𝐴)
mreexmrid.3 𝐼 = (mrInd‘𝐴)
mreexmrid.4 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
mreexmrid.5 (𝜑𝑆𝐼)
mreexmrid.6 (𝜑𝑌𝑋)
mreexmrid.7 (𝜑 → ¬ 𝑌 ∈ (𝑁𝑆))
Assertion
Ref Expression
mreexmrid (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)
Distinct variable groups:   𝑋,𝑠,𝑦   𝑆,𝑠,𝑧,𝑦   𝜑,𝑠,𝑦,𝑧   𝑌,𝑠,𝑦,𝑧   𝑁,𝑠,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑠)   𝐼(𝑦,𝑧,𝑠)   𝑋(𝑧)

Proof of Theorem mreexmrid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mreexmrid.2 . 2 𝑁 = (mrCls‘𝐴)
2 mreexmrid.3 . 2 𝐼 = (mrInd‘𝐴)
3 mreexmrid.1 . 2 (𝜑𝐴 ∈ (Moore‘𝑋))
4 mreexmrid.5 . . . 4 (𝜑𝑆𝐼)
52, 3, 4mrissd 16498 . . 3 (𝜑𝑆𝑋)
6 mreexmrid.6 . . . 4 (𝜑𝑌𝑋)
76snssd 4485 . . 3 (𝜑 → {𝑌} ⊆ 𝑋)
85, 7unssd 3932 . 2 (𝜑 → (𝑆 ∪ {𝑌}) ⊆ 𝑋)
933ad2ant1 1128 . . . . . . . . . 10 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝐴 ∈ (Moore‘𝑋))
109elfvexd 6383 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑋 ∈ V)
11 mreexmrid.4 . . . . . . . . . 10 (𝜑 → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
12113ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ∀𝑠 ∈ 𝒫 𝑋𝑦𝑋𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))
1343ad2ant1 1128 . . . . . . . . . . 11 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑆𝐼)
142, 9, 13mrissd 16498 . . . . . . . . . 10 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑆𝑋)
1514ssdifssd 3891 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → (𝑆 ∖ {𝑥}) ⊆ 𝑋)
1663ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑌𝑋)
17 simp3 1133 . . . . . . . . . 10 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
18 difundir 4023 . . . . . . . . . . . 12 ((𝑆 ∪ {𝑌}) ∖ {𝑥}) = ((𝑆 ∖ {𝑥}) ∪ ({𝑌} ∖ {𝑥}))
19 simp2 1132 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑥𝑆)
203, 1, 5mrcssidd 16487 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ⊆ (𝑁𝑆))
21 mreexmrid.7 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ 𝑌 ∈ (𝑁𝑆))
2220, 21ssneldd 3747 . . . . . . . . . . . . . . . . 17 (𝜑 → ¬ 𝑌𝑆)
23223ad2ant1 1128 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑌𝑆)
24 nelneq 2863 . . . . . . . . . . . . . . . 16 ((𝑥𝑆 ∧ ¬ 𝑌𝑆) → ¬ 𝑥 = 𝑌)
2519, 23, 24syl2anc 696 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑥 = 𝑌)
26 elsni 4338 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑌} → 𝑥 = 𝑌)
2725, 26nsyl 135 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑥 ∈ {𝑌})
28 difsnb 4482 . . . . . . . . . . . . . 14 𝑥 ∈ {𝑌} ↔ ({𝑌} ∖ {𝑥}) = {𝑌})
2927, 28sylib 208 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ({𝑌} ∖ {𝑥}) = {𝑌})
3029uneq2d 3910 . . . . . . . . . . . 12 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ((𝑆 ∖ {𝑥}) ∪ ({𝑌} ∖ {𝑥})) = ((𝑆 ∖ {𝑥}) ∪ {𝑌}))
3118, 30syl5eq 2806 . . . . . . . . . . 11 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ((𝑆 ∪ {𝑌}) ∖ {𝑥}) = ((𝑆 ∖ {𝑥}) ∪ {𝑌}))
3231fveq2d 6356 . . . . . . . . . 10 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})) = (𝑁‘((𝑆 ∖ {𝑥}) ∪ {𝑌})))
3317, 32eleqtrd 2841 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑥 ∈ (𝑁‘((𝑆 ∖ {𝑥}) ∪ {𝑌})))
341, 2, 9, 13, 19ismri2dad 16499 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑥 ∈ (𝑁‘(𝑆 ∖ {𝑥})))
3510, 12, 15, 16, 33, 34mreexd 16504 . . . . . . . 8 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → 𝑌 ∈ (𝑁‘((𝑆 ∖ {𝑥}) ∪ {𝑥})))
36213ad2ant1 1128 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑌 ∈ (𝑁𝑆))
37 undif1 4187 . . . . . . . . . . 11 ((𝑆 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})
3819snssd 4485 . . . . . . . . . . . 12 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → {𝑥} ⊆ 𝑆)
39 ssequn2 3929 . . . . . . . . . . . 12 ({𝑥} ⊆ 𝑆 ↔ (𝑆 ∪ {𝑥}) = 𝑆)
4038, 39sylib 208 . . . . . . . . . . 11 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → (𝑆 ∪ {𝑥}) = 𝑆)
4137, 40syl5eq 2806 . . . . . . . . . 10 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ((𝑆 ∖ {𝑥}) ∪ {𝑥}) = 𝑆)
4241fveq2d 6356 . . . . . . . . 9 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → (𝑁‘((𝑆 ∖ {𝑥}) ∪ {𝑥})) = (𝑁𝑆))
4336, 42neleqtrrd 2861 . . . . . . . 8 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) → ¬ 𝑌 ∈ (𝑁‘((𝑆 ∖ {𝑥}) ∪ {𝑥})))
4435, 43pm2.65i 185 . . . . . . 7 ¬ (𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
45 df-3an 1074 . . . . . . 7 ((𝜑𝑥𝑆𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))) ↔ ((𝜑𝑥𝑆) ∧ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥}))))
4644, 45mtbi 311 . . . . . 6 ¬ ((𝜑𝑥𝑆) ∧ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
4746imnani 438 . . . . 5 ((𝜑𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
4847adantlr 753 . . . 4 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥𝑆) → ¬ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
4926adantl 473 . . . . . 6 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → 𝑥 = 𝑌)
5021ad2antrr 764 . . . . . 6 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ¬ 𝑌 ∈ (𝑁𝑆))
5149, 50eqneltrd 2858 . . . . 5 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ¬ 𝑥 ∈ (𝑁𝑆))
5249sneqd 4333 . . . . . . . . 9 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → {𝑥} = {𝑌})
5352difeq2d 3871 . . . . . . . 8 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ((𝑆 ∪ {𝑌}) ∖ {𝑥}) = ((𝑆 ∪ {𝑌}) ∖ {𝑌}))
54 difun2 4192 . . . . . . . 8 ((𝑆 ∪ {𝑌}) ∖ {𝑌}) = (𝑆 ∖ {𝑌})
5553, 54syl6eq 2810 . . . . . . 7 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ((𝑆 ∪ {𝑌}) ∖ {𝑥}) = (𝑆 ∖ {𝑌}))
56 difsnb 4482 . . . . . . . . 9 𝑌𝑆 ↔ (𝑆 ∖ {𝑌}) = 𝑆)
5722, 56sylib 208 . . . . . . . 8 (𝜑 → (𝑆 ∖ {𝑌}) = 𝑆)
5857ad2antrr 764 . . . . . . 7 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → (𝑆 ∖ {𝑌}) = 𝑆)
5955, 58eqtrd 2794 . . . . . 6 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ((𝑆 ∪ {𝑌}) ∖ {𝑥}) = 𝑆)
6059fveq2d 6356 . . . . 5 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})) = (𝑁𝑆))
6151, 60neleqtrrd 2861 . . . 4 (((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) ∧ 𝑥 ∈ {𝑌}) → ¬ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
62 simpr 479 . . . . 5 ((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) → 𝑥 ∈ (𝑆 ∪ {𝑌}))
63 elun 3896 . . . . 5 (𝑥 ∈ (𝑆 ∪ {𝑌}) ↔ (𝑥𝑆𝑥 ∈ {𝑌}))
6462, 63sylib 208 . . . 4 ((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) → (𝑥𝑆𝑥 ∈ {𝑌}))
6548, 61, 64mpjaodan 862 . . 3 ((𝜑𝑥 ∈ (𝑆 ∪ {𝑌})) → ¬ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
6665ralrimiva 3104 . 2 (𝜑 → ∀𝑥 ∈ (𝑆 ∪ {𝑌}) ¬ 𝑥 ∈ (𝑁‘((𝑆 ∪ {𝑌}) ∖ {𝑥})))
671, 2, 3, 8, 66ismri2dd 16496 1 (𝜑 → (𝑆 ∪ {𝑌}) ∈ 𝐼)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  cdif 3712  cun 3713  wss 3715  𝒫 cpw 4302  {csn 4321  cfv 6049  Moorecmre 16444  mrClscmrc 16445  mrIndcmri 16446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-mre 16448  df-mrc 16449  df-mri 16450
This theorem is referenced by:  mreexexlem2d  16507
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