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Theorem filssufilg 23898
Description: A filter is contained in some ultrafilter. This version of filssufil 23899 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 483 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2 rabss 4067 . . . . 5 ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
3 filsspw 23838 . . . . . . 7 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4 velpw 4611 . . . . . . 7 (𝑔 ∈ 𝒫 𝒫 𝑋𝑔 ⊆ 𝒫 𝑋)
53, 4sylibr 233 . . . . . 6 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
65a1d 25 . . . . 5 (𝑔 ∈ (Fil‘𝑋) → (𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
72, 6mprgbir 3057 . . . 4 {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋
8 ssnum 10078 . . . 4 ((𝒫 𝒫 𝑋 ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
91, 7, 8sylancl 584 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
10 ssid 4001 . . . . . . 7 𝐹𝐹
1110jctr 523 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
12 sseq2 4005 . . . . . . 7 (𝑔 = 𝐹 → (𝐹𝑔𝐹𝐹))
1312elrab 3680 . . . . . 6 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
1411, 13sylibr 233 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
1514ne0d 4337 . . . 4 (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
1615adantr 479 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
17 sseq2 4005 . . . . . . 7 (𝑔 = 𝑥 → (𝐹𝑔𝐹 𝑥))
18 simpr1 1191 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
19 ssrab 4068 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2018, 19sylib 217 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2120simpld 493 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22 simpr2 1192 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ≠ ∅)
23 simpr3 1193 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → [] Or 𝑥)
24 sorpssun 7740 . . . . . . . . . 10 (( [] Or 𝑥 ∧ (𝑔𝑥𝑥)) → (𝑔) ∈ 𝑥)
2524ralrimivva 3190 . . . . . . . . 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 23872 . . . . . . . 8 ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥) → 𝑥 ∈ (Fil‘𝑋))
2821, 22, 26, 27syl3anc 1368 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ (Fil‘𝑋))
29 n0 4348 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ ∃ 𝑥)
30 ssel2 3973 . . . . . . . . . . . . . 14 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
31 sseq2 4005 . . . . . . . . . . . . . . 15 (𝑔 = → (𝐹𝑔𝐹))
3231elrab 3680 . . . . . . . . . . . . . 14 ( ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( ∈ (Fil‘𝑋) ∧ 𝐹))
3330, 32sylib 217 . . . . . . . . . . . . 13 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ( ∈ (Fil‘𝑋) ∧ 𝐹))
3433simprd 494 . . . . . . . . . . . 12 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹)
35 ssuni 4939 . . . . . . . . . . . 12 ((𝐹𝑥) → 𝐹 𝑥)
3634, 35sylancom 586 . . . . . . . . . . 11 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹 𝑥)
3736ex 411 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥𝐹 𝑥))
3837exlimdv 1928 . . . . . . . . 9 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (∃ 𝑥𝐹 𝑥))
3929, 38biimtrid 241 . . . . . . . 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 3682 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
4241ex 411 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4342alrimiv 1922 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4443adantr 479 . . 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 10544 . . 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 1368 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
47 sseq2 4005 . . . . 5 (𝑔 = 𝑓 → (𝐹𝑔𝐹𝑓))
4847elrab 3680 . . . 4 (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓))
4931ralrab 3686 . . . 4 (∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 ↔ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓))
50 simpll 765 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
51 sstr2 3985 . . . . . . . . . . 11 (𝐹𝑓 → (𝑓𝐹))
5251imim1d 82 . . . . . . . . . 10 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓 → ¬ 𝑓)))
53 df-pss 3966 . . . . . . . . . . . . 13 (𝑓 ↔ (𝑓𝑓))
5453simplbi2 499 . . . . . . . . . . . 12 (𝑓 → (𝑓𝑓))
5554necon1bd 2947 . . . . . . . . . . 11 (𝑓 → (¬ 𝑓𝑓 = ))
5655a2i 14 . . . . . . . . . 10 ((𝑓 → ¬ 𝑓) → (𝑓𝑓 = ))
5752, 56syl6 35 . . . . . . . . 9 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓𝑓 = )))
5857ralimdv 3158 . . . . . . . 8 (𝐹𝑓 → (∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
5958imp 405 . . . . . . 7 ((𝐹𝑓 ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
6059adantll 712 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
61 isufil2 23895 . . . . . 6 (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6250, 60, 61sylanbrc 581 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (UFil‘𝑋))
63 simplr 767 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝐹𝑓)
6462, 63jca 510 . . . 4 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6548, 49, 64syl2anb 596 . . 3 ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ ∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6665reximi2 3068 . 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 394  w3a 1084  wal 1531  wex 1773  wcel 2098  wne 2929  wral 3050  wrex 3059  {crab 3418  cun 3944  wss 3946  wpss 3947  c0 4324  𝒫 cpw 4606   cuni 4912   Or wor 5592  dom cdm 5681  cfv 6553   [] crpss 7732  cardccrd 9974  Filcfil 23832  UFilcufil 23886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-se 5637  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-pred 6311  df-ord 6378  df-on 6379  df-lim 6380  df-suc 6381  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-rpss 7733  df-om 7876  df-1st 8002  df-2nd 8003  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-2o 8496  df-oadd 8499  df-er 8733  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9450  df-dju 9940  df-card 9978  df-fbas 21332  df-fg 21333  df-fil 23833  df-ufil 23888
This theorem is referenced by:  filssufil  23899  numufl  23902
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