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Theorem ufilen 22222
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 8363 . . . . . 6 Rel ≼
21brrelex2i 5495 . . . . 5 (ω ≼ 𝑋𝑋 ∈ V)
3 numth3 9738 . . . . 5 (𝑋 ∈ V → 𝑋 ∈ dom card)
42, 3syl 17 . . . 4 (ω ≼ 𝑋𝑋 ∈ dom card)
5 csdfil 22186 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ∈ (Fil‘𝑋))
64, 5mpancom 684 . . 3 (ω ≼ 𝑋 → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ∈ (Fil‘𝑋))
7 filssufil 22204 . . 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 6571 . . . . . . 7 (𝑓 ∈ (UFil‘𝑋) → 𝑋 ∈ V)
109ad2antlr 723 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑋 ∈ V)
11 ufilfil 22196 . . . . . . . 8 (𝑓 ∈ (UFil‘𝑋) → 𝑓 ∈ (Fil‘𝑋))
12 filelss 22144 . . . . . . . 8 ((𝑓 ∈ (Fil‘𝑋) ∧ 𝑥𝑓) → 𝑥𝑋)
1311, 12sylan 580 . . . . . . 7 ((𝑓 ∈ (UFil‘𝑋) ∧ 𝑥𝑓) → 𝑥𝑋)
1413adantll 710 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑥𝑋)
15 ssdomg 8403 . . . . . 6 (𝑋 ∈ V → (𝑥𝑋𝑥𝑋))
1610, 14, 15sylc 65 . . . . 5 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → 𝑥𝑋)
17 filfbas 22140 . . . . . . . . 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 22131 . . . . . . 7 ((𝑓 ∈ (fBas‘𝑋) ∧ 𝑥𝑓) → ¬ (𝑋𝑥) ∈ 𝑓)
2119, 20sylan 580 . . . . . 6 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ¬ (𝑋𝑥) ∈ 𝑓)
22 difeq2 4014 . . . . . . . . . . . . 13 (𝑦 = (𝑋𝑥) → (𝑋𝑦) = (𝑋 ∖ (𝑋𝑥)))
2322breq1d 4972 . . . . . . . . . . . 12 (𝑦 = (𝑋𝑥) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) ≺ 𝑋))
24 difss 4029 . . . . . . . . . . . . . 14 (𝑋𝑥) ⊆ 𝑋
25 elpw2g 5138 . . . . . . . . . . . . . 14 (𝑋 ∈ V → ((𝑋𝑥) ∈ 𝒫 𝑋 ↔ (𝑋𝑥) ⊆ 𝑋))
2624, 25mpbiri 259 . . . . . . . . . . . . 13 (𝑋 ∈ V → (𝑋𝑥) ∈ 𝒫 𝑋)
27263ad2ant1 1126 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋𝑥) ∈ 𝒫 𝑋)
28 simp2 1130 . . . . . . . . . . . . . 14 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → 𝑥𝑋)
29 dfss4 4155 . . . . . . . . . . . . . 14 (𝑥𝑋 ↔ (𝑋 ∖ (𝑋𝑥)) = 𝑥)
3028, 29sylib 219 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋 ∖ (𝑋𝑥)) = 𝑥)
31 simp3 1131 . . . . . . . . . . . . 13 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → 𝑥𝑋)
3230, 31eqbrtrd 4984 . . . . . . . . . . . 12 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋 ∖ (𝑋𝑥)) ≺ 𝑋)
3323, 27, 32elrabd 3620 . . . . . . . . . . 11 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → (𝑋𝑥) ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋})
34 ssel 3883 . . . . . . . . . . 11 ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ((𝑋𝑥) ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} → (𝑋𝑥) ∈ 𝑓))
3533, 34syl5com 31 . . . . . . . . . 10 ((𝑋 ∈ V ∧ 𝑥𝑋𝑥𝑋) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → (𝑋𝑥) ∈ 𝑓))
36353expa 1111 . . . . . . . . 9 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ 𝑥𝑋) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → (𝑋𝑥) ∈ 𝑓))
3736impancom 452 . . . . . . . 8 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓) → (𝑥𝑋 → (𝑋𝑥) ∈ 𝑓))
3837con3d 155 . . . . . . 7 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓) → (¬ (𝑋𝑥) ∈ 𝑓 → ¬ 𝑥𝑋))
3938impancom 452 . . . . . 6 (((𝑋 ∈ V ∧ 𝑥𝑋) ∧ ¬ (𝑋𝑥) ∈ 𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ¬ 𝑥𝑋))
4010, 14, 21, 39syl21anc 834 . . . . 5 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ¬ 𝑥𝑋))
41 bren2 8388 . . . . . 6 (𝑥𝑋 ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑋))
4241simplbi2 501 . . . . 5 (𝑥𝑋 → (¬ 𝑥𝑋𝑥𝑋))
4316, 40, 42sylsyld 61 . . . 4 (((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) ∧ 𝑥𝑓) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓𝑥𝑋))
4443ralrimdva 3156 . . 3 ((ω ≼ 𝑋𝑓 ∈ (UFil‘𝑋)) → ({𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ∀𝑥𝑓 𝑥𝑋))
4544reximdva 3237 . 2 (ω ≼ 𝑋 → (∃𝑓 ∈ (UFil‘𝑋){𝑦 ∈ 𝒫 𝑋 ∣ (𝑋𝑦) ≺ 𝑋} ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋))
468, 45mpd 15 1 (ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥𝑓 𝑥𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  cdif 3856  wss 3859  𝒫 cpw 4453   class class class wbr 4962  dom cdm 5443  cfv 6225  ωcom 7436  cen 8354  cdom 8355  csdm 8356  cardccrd 9210  fBascfbas 20215  Filcfil 22137  UFilcufil 22191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-ac2 9731
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-rpss 7307  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-oi 8820  df-dju 9176  df-card 9214  df-ac 9388  df-fbas 20224  df-fg 20225  df-fil 22138  df-ufil 22193
This theorem is referenced by: (None)
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