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Theorem filssufilg 21638
Description: A filter is contained in some ultrafilter. This version of filssufil 21639 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 477 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → 𝒫 𝒫 𝑋 ∈ dom card)
2 rabss 3663 . . . . 5 ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
3 filsspw 21578 . . . . . . 7 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4 selpw 4142 . . . . . . 7 (𝑔 ∈ 𝒫 𝒫 𝑋𝑔 ⊆ 𝒫 𝑋)
53, 4sylibr 224 . . . . . 6 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
65a1d 25 . . . . 5 (𝑔 ∈ (Fil‘𝑋) → (𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
72, 6mprgbir 2922 . . . 4 {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋
8 ssnum 8814 . . . 4 ((𝒫 𝒫 𝑋 ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
91, 7, 8sylancl 693 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
10 ssid 3608 . . . . . . 7 𝐹𝐹
1110jctr 564 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
12 sseq2 3611 . . . . . . 7 (𝑔 = 𝐹 → (𝐹𝑔𝐹𝐹))
1312elrab 3350 . . . . . 6 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
1411, 13sylibr 224 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
15 ne0i 3902 . . . . 5 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
1614, 15syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
1716adantr 481 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
18 simpr1 1065 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
19 ssrab 3664 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2018, 19sylib 208 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2120simpld 475 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
22 simpr2 1066 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ≠ ∅)
23 simpr3 1067 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → [] Or 𝑥)
24 sorpssun 6904 . . . . . . . . . 10 (( [] Or 𝑥 ∧ (𝑔𝑥𝑥)) → (𝑔) ∈ 𝑥)
2524ralrimivva 2966 . . . . . . . . 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 21612 . . . . . . . 8 ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥) → 𝑥 ∈ (Fil‘𝑋))
2821, 22, 26, 27syl3anc 1323 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ (Fil‘𝑋))
29 n0 3912 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ ∃ 𝑥)
30 ssel2 3582 . . . . . . . . . . . . . 14 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
31 sseq2 3611 . . . . . . . . . . . . . . 15 (𝑔 = → (𝐹𝑔𝐹))
3231elrab 3350 . . . . . . . . . . . . . 14 ( ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( ∈ (Fil‘𝑋) ∧ 𝐹))
3330, 32sylib 208 . . . . . . . . . . . . 13 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ( ∈ (Fil‘𝑋) ∧ 𝐹))
3433simprd 479 . . . . . . . . . . . 12 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹)
35 ssuni 4430 . . . . . . . . . . . 12 ((𝐹𝑥) → 𝐹 𝑥)
3634, 35sylancom 700 . . . . . . . . . . 11 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹 𝑥)
3736ex 450 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥𝐹 𝑥))
3837exlimdv 1858 . . . . . . . . 9 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (∃ 𝑥𝐹 𝑥))
3929, 38syl5bi 232 . . . . . . . 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)
41 sseq2 3611 . . . . . . . 8 (𝑔 = 𝑥 → (𝐹𝑔𝐹 𝑥))
4241elrab 3350 . . . . . . 7 ( 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( 𝑥 ∈ (Fil‘𝑋) ∧ 𝐹 𝑥))
4328, 40, 42sylanbrc 697 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
4443ex 450 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4544alrimiv 1852 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4645adantr 481 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
47 zornn0g 9279 . . 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)
489, 17, 46, 47syl3anc 1323 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
49 sseq2 3611 . . . . 5 (𝑔 = 𝑓 → (𝐹𝑔𝐹𝑓))
5049elrab 3350 . . . 4 (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓))
5131ralrab 3354 . . . 4 (∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 ↔ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓))
52 simpll 789 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
53 sstr2 3594 . . . . . . . . . . 11 (𝐹𝑓 → (𝑓𝐹))
5453imim1d 82 . . . . . . . . . 10 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓 → ¬ 𝑓)))
55 df-pss 3575 . . . . . . . . . . . . 13 (𝑓 ↔ (𝑓𝑓))
5655simplbi2 654 . . . . . . . . . . . 12 (𝑓 → (𝑓𝑓))
5756necon1bd 2808 . . . . . . . . . . 11 (𝑓 → (¬ 𝑓𝑓 = ))
5857a2i 14 . . . . . . . . . 10 ((𝑓 → ¬ 𝑓) → (𝑓𝑓 = ))
5954, 58syl6 35 . . . . . . . . 9 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓𝑓 = )))
6059ralimdv 2958 . . . . . . . 8 (𝐹𝑓 → (∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6160imp 445 . . . . . . 7 ((𝐹𝑓 ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
6261adantll 749 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
63 isufil2 21635 . . . . . 6 (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6452, 62, 63sylanbrc 697 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (UFil‘𝑋))
65 simplr 791 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝐹𝑓)
6664, 65jca 554 . . . 4 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6750, 51, 66syl2anb 496 . . 3 ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ ∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6867reximi2 3005 . 2 (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
6948, 68syl 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 384  w3a 1036  wal 1478  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cun 3557  wss 3559  wpss 3560  c0 3896  𝒫 cpw 4135   cuni 4407   Or wor 4999  dom cdm 5079  cfv 5852   [] crpss 6896  cardccrd 8713  Filcfil 21572  UFilcufil 21626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-rpss 6897  df-om 7020  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-fin 7911  df-fi 8269  df-card 8717  df-cda 8942  df-fbas 19675  df-fg 19676  df-fil 21573  df-ufil 21628
This theorem is referenced by:  filssufil  21639  numufl  21642
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