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Theorem ufilen 21715
Description: Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.)
Assertion
Ref Expression
ufilen (ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋)
Distinct variable group:   𝑥,𝑓,𝑋

Proof of Theorem ufilen
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 reldom 7946 . . . . . 6 Rel ≼
21brrelex2i 5149 . . . . 5 (ω ≼ 𝑋𝑋 ∈ V)
3 numth3 9277 . . . . 5 (𝑋 ∈ V → 𝑋 ∈ dom card)
42, 3syl 17 . . . 4 (ω ≼ 𝑋𝑋 ∈ dom card)
5 csdfil 21679 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ∈ (Fil‘𝑋))
64, 5mpancom 702 . . 3 (ω ≼ 𝑋 → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ∈ (Fil‘𝑋))
7 filssufil 21697 . . 3 ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋){𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓)
86, 7syl 17 . 2 (ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋){𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓)
9 elfvex 6208 . . . . . . 7 (𝑓 ∈ (UFil‘𝑋) → 𝑋 ∈ V)
109ad2antlr 762 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑋 ∈ V)
11 ufilfil 21689 . . . . . . . 8 (𝑓 ∈ (UFil‘𝑋) → 𝑓 ∈ (Fil‘𝑋))
12 filelss 21637 . . . . . . . 8 ((𝑓 ∈ (Fil‘𝑋) ∧ 𝑥𝑓) → 𝑥𝑋)
1311, 12sylan 488 . . . . . . 7 ((𝑓 ∈ (UFil‘𝑋) ∧ 𝑥𝑓) → 𝑥𝑋)
1413adantll 749 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑥𝑋)
15 ssdomg 7986 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋𝑥𝑋))
1610, 14, 15sylc 65 . . . . 5 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑥𝑋)
17 filfbas 21633 . . . . . . . . 9 (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
1811, 17syl 17 . . . . . . . 8 (𝑓 ∈ (UFil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
1918adantl 482 . . . . . . 7 ((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) → 𝑓 ∈ (fBas‘𝑋))
20 fbncp 21624 . . . . . . 7 ((𝑓 ∈ (fBas‘𝑋) ∧ 𝑥𝑓) → ¬ (𝑋𝑥) ∈ 𝑓)
2119, 20sylan 488 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ¬ (𝑋𝑥) ∈ 𝑓)
22 difss 3729 . . . . . . . . . . . . . 14 (𝑋𝑥) ⊆ 𝑋
23 elpw2g 4818 . . . . . . . . . . . . . 14 (𝑋 ∈ V → ((𝑋𝑥) ∈ 𝒫 𝑋 ↔ (𝑋𝑥) ⊆ 𝑋))
2422, 23mpbiri 248 . . . . . . . . . . . . 13 (𝑋 ∈ V → (𝑋𝑥) ∈ 𝒫 𝑋)
25243ad2ant1 1080 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋𝑥) ∈ 𝒫 𝑋)
26 simp2 1060 . . . . . . . . . . . . . 14 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → 𝑥𝑋)
27 dfss4 3850 . . . . . . . . . . . . . 14 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
2826, 27sylib 208 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
29 simp3 1061 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → 𝑥𝑋)
3028, 29eqbrtrd 4666 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋 ∖ (𝑋𝑥)) ≺ 𝑋)
31 difeq2 3714 . . . . . . . . . . . . . 14 (𝑦 = (𝑋𝑥) → (𝑋𝑦) = (𝑋 ∖ (𝑋𝑥)))
3231breq1d 4654 . . . . . . . . . . . . 13 (𝑦 = (𝑋𝑥) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) ≺ 𝑋))
3332elrab 3357 . . . . . . . . . . . 12 ((𝑋𝑥) ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ↔ ((𝑋𝑥) ∈ 𝒫 𝑋 ∧ (𝑋 ∖ (𝑋𝑥)) ≺ 𝑋))
3425, 30, 33sylanbrc 697 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋𝑥) ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋})
35 ssel 3589 . . . . . . . . . . 11 ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ((𝑋𝑥) ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} → (𝑋𝑥) ∈ 𝑓))
3634, 35syl5com 31 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → (𝑋𝑥) ∈ 𝑓))
37363expa 1263 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ 𝑥𝑋) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → (𝑋𝑥) ∈ 𝑓))
3837impancom 456 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓) → (𝑥𝑋 → (𝑋𝑥) ∈ 𝑓))
3938con3d 148 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓) → (¬ (𝑋𝑥) ∈ 𝑓 → ¬ 𝑥𝑋))
4039impancom 456 . . . . . 6 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ ¬ (𝑋𝑥) ∈ 𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ¬ 𝑥𝑋))
4110, 14, 21, 40syl21anc 1323 . . . . 5 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ¬ 𝑥𝑋))
42 bren2 7971 . . . . . 6 (𝑥𝑋 ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑋))
4342simplbi2 654 . . . . 5 (𝑥𝑋 → (¬ 𝑥𝑋𝑥𝑋))
4416, 41, 43sylsyld 61 . . . 4 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓𝑥𝑋))
4544ralrimdva 2966 . . 3 ((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ∀𝑥𝑓 𝑥𝑋))
4645reximdva 3014 . 2 (ω ≼ 𝑋 → (∃𝑓 ∈ (UFil‘𝑋){𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋))
478, 46mpd 15 1 (ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  wrex 2910  {crab 2913  Vcvv 3195  cdif 3564  wss 3567  𝒫 cpw 4149   class class class wbr 4644  dom cdm 5104  cfv 5876  ωcom 7050  cen 7937  cdom 7938  csdm 7939  cardccrd 8746  fBascfbas 19715  Filcfil 21630  UFilcufil 21684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-rpss 6922  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-2o 7546  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-fi 8302  df-oi 8400  df-card 8750  df-ac 8924  df-cda 8975  df-fbas 19724  df-fg 19725  df-fil 21631  df-ufil 21686
This theorem is referenced by: (None)
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