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Theorem filssufilg 23935
Description: A filter is contained in some ultrafilter. This version of filssufil 23936 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 4082 . . . . 5 ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
3 filsspw 23875 . . . . . . 7 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4 velpw 4610 . . . . . . 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 3066 . . . 4 {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋
8 ssnum 10077 . . . 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 4018 . . . . . . 7 𝐹𝐹
1110jctr 524 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
12 sseq2 4022 . . . . . . 7 (𝑔 = 𝐹 → (𝐹𝑔𝐹𝐹))
1312elrab 3695 . . . . . 6 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
1411, 13sylibr 234 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
1514ne0d 4348 . . . 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 4022 . . . . . . 7 (𝑔 = 𝑥 → (𝐹𝑔𝐹 𝑥))
18 simpr1 1193 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
19 ssrab 4083 . . . . . . . . . 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 1194 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ≠ ∅)
23 simpr3 1195 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → [] Or 𝑥)
24 sorpssun 7749 . . . . . . . . . 10 (( [] Or 𝑥 ∧ (𝑔𝑥𝑥)) → (𝑔) ∈ 𝑥)
2524ralrimivva 3200 . . . . . . . . 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 23909 . . . . . . . 8 ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥) → 𝑥 ∈ (Fil‘𝑋))
2821, 22, 26, 27syl3anc 1370 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ (Fil‘𝑋))
29 n0 4359 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ ∃ 𝑥)
30 ssel2 3990 . . . . . . . . . . . . . 14 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
31 sseq2 4022 . . . . . . . . . . . . . . 15 (𝑔 = → (𝐹𝑔𝐹))
3231elrab 3695 . . . . . . . . . . . . . 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 4937 . . . . . . . . . . . 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 1931 . . . . . . . . 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 3697 . . . . . 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 1925 . . . 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 10543 . . 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 1370 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
47 sseq2 4022 . . . . 5 (𝑔 = 𝑓 → (𝐹𝑔𝐹𝑓))
4847elrab 3695 . . . 4 (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓))
4931ralrab 3702 . . . 4 (∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 ↔ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓))
50 simpll 767 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
51 sstr2 4002 . . . . . . . . . . 11 (𝐹𝑓 → (𝑓𝐹))
5251imim1d 82 . . . . . . . . . 10 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓 → ¬ 𝑓)))
53 df-pss 3983 . . . . . . . . . . . . 13 (𝑓 ↔ (𝑓𝑓))
5453simplbi2 500 . . . . . . . . . . . 12 (𝑓 → (𝑓𝑓))
5554necon1bd 2956 . . . . . . . . . . 11 (𝑓 → (¬ 𝑓𝑓 = ))
5655a2i 14 . . . . . . . . . 10 ((𝑓 → ¬ 𝑓) → (𝑓𝑓 = ))
5752, 56syl6 35 . . . . . . . . 9 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓𝑓 = )))
5857ralimdv 3167 . . . . . . . 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 23932 . . . . . 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 769 . . . . 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 3077 . 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 1535  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  cun 3961  wss 3963  wpss 3964  c0 4339  𝒫 cpw 4605   cuni 4912   Or wor 5596  dom cdm 5689  cfv 6563   [] crpss 7741  cardccrd 9973  Filcfil 23869  UFilcufil 23923
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-dju 9939  df-card 9977  df-fbas 21379  df-fg 21380  df-fil 23870  df-ufil 23925
This theorem is referenced by:  filssufil  23936  numufl  23939
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