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Theorem filssufilg 23920
Description: A filter is contained in some ultrafilter. This version of filssufil 23921 contains the choice as a hypothesis (in the assumption that 𝒫 𝒫 𝑋 is well-orderable). (Contributed by Mario Carneiro, 24-May-2015.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filssufilg ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Distinct variable groups:   𝑓,𝐹   𝑓,𝑋

Proof of Theorem filssufilg
Dummy variables 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2 rabss 4071 . . . . 5 ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
3 filsspw 23860 . . . . . . 7 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4 velpw 4604 . . . . . . 7 (𝑔 ∈ 𝒫 𝒫 𝑋𝑔 ⊆ 𝒫 𝑋)
53, 4sylibr 234 . . . . . 6 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
65a1d 25 . . . . 5 (𝑔 ∈ (Fil‘𝑋) → (𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
72, 6mprgbir 3067 . . . 4 {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋
8 ssnum 10080 . . . 4 ((𝒫 𝒫 𝑋 ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
91, 7, 8sylancl 586 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
10 ssid 4005 . . . . . . 7 𝐹𝐹
1110jctr 524 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
12 sseq2 4009 . . . . . . 7 (𝑔 = 𝐹 → (𝐹𝑔𝐹𝐹))
1312elrab 3691 . . . . . 6 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
1411, 13sylibr 234 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
1514ne0d 4341 . . . 4 (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
1615adantr 480 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
17 sseq2 4009 . . . . . . 7 (𝑔 = 𝑥 → (𝐹𝑔𝐹 𝑥))
18 simpr1 1194 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
19 ssrab 4072 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2018, 19sylib 218 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2120simpld 494 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22 simpr2 1195 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ≠ ∅)
23 simpr3 1196 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → [] Or 𝑥)
24 sorpssun 7751 . . . . . . . . . 10 (( [] Or 𝑥 ∧ (𝑔𝑥𝑥)) → (𝑔) ∈ 𝑥)
2524ralrimivva 3201 . . . . . . . . 9 ( [] Or 𝑥 → ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥)
2623, 25syl 17 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥)
27 filuni 23894 . . . . . . . 8 ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥) → 𝑥 ∈ (Fil‘𝑋))
2821, 22, 26, 27syl3anc 1372 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ (Fil‘𝑋))
29 n0 4352 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ ∃ 𝑥)
30 ssel2 3977 . . . . . . . . . . . . . 14 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
31 sseq2 4009 . . . . . . . . . . . . . . 15 (𝑔 = → (𝐹𝑔𝐹))
3231elrab 3691 . . . . . . . . . . . . . 14 ( ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( ∈ (Fil‘𝑋) ∧ 𝐹))
3330, 32sylib 218 . . . . . . . . . . . . 13 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ( ∈ (Fil‘𝑋) ∧ 𝐹))
3433simprd 495 . . . . . . . . . . . 12 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹)
35 ssuni 4931 . . . . . . . . . . . 12 ((𝐹𝑥) → 𝐹 𝑥)
3634, 35sylancom 588 . . . . . . . . . . 11 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹 𝑥)
3736ex 412 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥𝐹 𝑥))
3837exlimdv 1932 . . . . . . . . 9 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (∃ 𝑥𝐹 𝑥))
3929, 38biimtrid 242 . . . . . . . 8 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥 ≠ ∅ → 𝐹 𝑥))
4018, 22, 39sylc 65 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝐹 𝑥)
4117, 28, 40elrabd 3693 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
4241ex 412 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4342alrimiv 1926 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4443adantr 480 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
45 zornn0g 10546 . . 3 (({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅ ∧ ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
469, 16, 44, 45syl3anc 1372 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
47 sseq2 4009 . . . . 5 (𝑔 = 𝑓 → (𝐹𝑔𝐹𝑓))
4847elrab 3691 . . . 4 (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓))
4931ralrab 3698 . . . 4 (∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 ↔ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓))
50 simpll 766 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
51 sstr2 3989 . . . . . . . . . . 11 (𝐹𝑓 → (𝑓𝐹))
5251imim1d 82 . . . . . . . . . 10 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓 → ¬ 𝑓)))
53 df-pss 3970 . . . . . . . . . . . . 13 (𝑓 ↔ (𝑓𝑓))
5453simplbi2 500 . . . . . . . . . . . 12 (𝑓 → (𝑓𝑓))
5554necon1bd 2957 . . . . . . . . . . 11 (𝑓 → (¬ 𝑓𝑓 = ))
5655a2i 14 . . . . . . . . . 10 ((𝑓 → ¬ 𝑓) → (𝑓𝑓 = ))
5752, 56syl6 35 . . . . . . . . 9 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓𝑓 = )))
5857ralimdv 3168 . . . . . . . 8 (𝐹𝑓 → (∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
5958imp 406 . . . . . . 7 ((𝐹𝑓 ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
6059adantll 714 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
61 isufil2 23917 . . . . . 6 (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6250, 60, 61sylanbrc 583 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (UFil‘𝑋))
63 simplr 768 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝐹𝑓)
6462, 63jca 511 . . . 4 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6548, 49, 64syl2anb 598 . . 3 ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ ∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6665reximi2 3078 . 2 (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
6746, 66syl 17 1 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1086  wal 1537  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  cun 3948  wss 3950  wpss 3951  c0 4332  𝒫 cpw 4599   cuni 4906   Or wor 5590  dom cdm 5684  cfv 6560   [] crpss 7743  cardccrd 9976  Filcfil 23854  UFilcufil 23908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-en 8987  df-dom 8988  df-fin 8990  df-fi 9452  df-dju 9942  df-card 9980  df-fbas 21362  df-fg 21363  df-fil 23855  df-ufil 23910
This theorem is referenced by:  filssufil  23921  numufl  23924
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