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Theorem ufinffr 22543
Description: An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)
Assertion
Ref Expression
ufinffr ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴𝑓 𝑓 = ∅))
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑋

Proof of Theorem ufinffr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ominf 8729 . . . . 5 ¬ ω ∈ Fin
2 domfi 8738 . . . . . 6 ((𝐴 ∈ Fin ∧ ω ≼ 𝐴) → ω ∈ Fin)
32expcom 417 . . . . 5 (ω ≼ 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin))
41, 3mtoi 202 . . . 4 (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin)
5 cfinfil 22507 . . . 4 ((𝑋𝐵𝐴𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
64, 5syl3an3 1162 . . 3 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋))
7 filssufil 22526 . . 3 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓)
86, 7syl 17 . 2 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓)
9 difeq2 4079 . . . . . . . 8 (𝑥 = 𝐴 → (𝐴𝑥) = (𝐴𝐴))
10 difid 4313 . . . . . . . 8 (𝐴𝐴) = ∅
119, 10syl6eq 2875 . . . . . . 7 (𝑥 = 𝐴 → (𝐴𝑥) = ∅)
1211eleq1d 2900 . . . . . 6 (𝑥 = 𝐴 → ((𝐴𝑥) ∈ Fin ↔ ∅ ∈ Fin))
13 elpw2g 5234 . . . . . . . 8 (𝑋𝐵 → (𝐴 ∈ 𝒫 𝑋𝐴𝑋))
1413biimpar 481 . . . . . . 7 ((𝑋𝐵𝐴𝑋) → 𝐴 ∈ 𝒫 𝑋)
15143adant3 1129 . . . . . 6 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → 𝐴 ∈ 𝒫 𝑋)
16 0fin 8745 . . . . . . 7 ∅ ∈ Fin
1716a1i 11 . . . . . 6 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ∅ ∈ Fin)
1812, 15, 17elrabd 3668 . . . . 5 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin})
19 ssel 3946 . . . . 5 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 → (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} → 𝐴𝑓))
2018, 19syl5com 31 . . . 4 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓𝐴𝑓))
21 intss 4883 . . . . . 6 ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 𝑓 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin})
22 neldifsn 4709 . . . . . . . . . 10 ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
23 elinti 4871 . . . . . . . . . 10 (𝑦 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} → ((𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} → 𝑦 ∈ (𝐴 ∖ {𝑦})))
2422, 23mtoi 202 . . . . . . . . 9 (𝑦 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} → ¬ (𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin})
25 difeq2 4079 . . . . . . . . . . 11 (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
2625eleq1d 2900 . . . . . . . . . 10 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
27 simp2 1134 . . . . . . . . . . . 12 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → 𝐴𝑋)
2827ssdifssd 4105 . . . . . . . . . . 11 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ⊆ 𝑋)
29 elpw2g 5234 . . . . . . . . . . . 12 (𝑋𝐵 → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝑋))
30293ad2ant1 1130 . . . . . . . . . . 11 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝑋))
3128, 30mpbird 260 . . . . . . . . . 10 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋)
32 snfi 8592 . . . . . . . . . . . 12 {𝑦} ∈ Fin
33 eldif 3929 . . . . . . . . . . . . . . 15 (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∖ {𝑦})))
34 eldif 3929 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
3534notbii 323 . . . . . . . . . . . . . . . . 17 𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
36 iman 405 . . . . . . . . . . . . . . . . 17 ((𝑥𝐴𝑥 ∈ {𝑦}) ↔ ¬ (𝑥𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
3735, 36bitr4i 281 . . . . . . . . . . . . . . . 16 𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥𝐴𝑥 ∈ {𝑦}))
3837anbi2i 625 . . . . . . . . . . . . . . 15 ((𝑥𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∖ {𝑦})) ↔ (𝑥𝐴 ∧ (𝑥𝐴𝑥 ∈ {𝑦})))
3933, 38bitri 278 . . . . . . . . . . . . . 14 (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) ↔ (𝑥𝐴 ∧ (𝑥𝐴𝑥 ∈ {𝑦})))
40 pm3.35 802 . . . . . . . . . . . . . 14 ((𝑥𝐴 ∧ (𝑥𝐴𝑥 ∈ {𝑦})) → 𝑥 ∈ {𝑦})
4139, 40sylbi 220 . . . . . . . . . . . . 13 (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) → 𝑥 ∈ {𝑦})
4241ssriv 3957 . . . . . . . . . . . 12 (𝐴 ∖ (𝐴 ∖ {𝑦})) ⊆ {𝑦}
43 ssfi 8737 . . . . . . . . . . . 12 (({𝑦} ∈ Fin ∧ (𝐴 ∖ (𝐴 ∖ {𝑦})) ⊆ {𝑦}) → (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin)
4432, 42, 43mp2an 691 . . . . . . . . . . 11 (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
4544a1i 11 . . . . . . . . . 10 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin)
4626, 31, 45elrabd 3668 . . . . . . . . 9 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin})
4724, 46nsyl3 140 . . . . . . . 8 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ¬ 𝑦 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin})
4847eq0rdv 4340 . . . . . . 7 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} = ∅)
4948sseq2d 3985 . . . . . 6 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ( 𝑓 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ↔ 𝑓 ⊆ ∅))
5021, 49syl5ib 247 . . . . 5 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 𝑓 ⊆ ∅))
51 ss0 4335 . . . . 5 ( 𝑓 ⊆ ∅ → 𝑓 = ∅)
5250, 51syl6 35 . . . 4 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 𝑓 = ∅))
5320, 52jcad 516 . . 3 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 → (𝐴𝑓 𝑓 = ∅)))
5453reximdv 3265 . 2 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → (∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥) ∈ Fin} ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)(𝐴𝑓 𝑓 = ∅)))
558, 54mpd 15 1 ((𝑋𝐵𝐴𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴𝑓 𝑓 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  {crab 3137  cdif 3916  wss 3919  c0 4276  𝒫 cpw 4522  {csn 4550   cint 4862   class class class wbr 5053  cfv 6345  ωcom 7576  cdom 8505  Fincfn 8507  Filcfil 22459  UFilcufil 22513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-ac2 9885
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-rpss 7445  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-fi 8874  df-dju 9329  df-card 9367  df-ac 9542  df-fbas 20097  df-fg 20098  df-fil 22460  df-ufil 22515
This theorem is referenced by: (None)
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