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Theorem filssufilg 21994
Description: A filter is contained in some ultrafilter. This version of filssufil 21995 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 3839 . . . . 5 ({𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋 ↔ ∀𝑔 ∈ (Fil‘𝑋)(𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
3 filsspw 21934 . . . . . . 7 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ⊆ 𝒫 𝑋)
4 selpw 4322 . . . . . . 7 (𝑔 ∈ 𝒫 𝒫 𝑋𝑔 ⊆ 𝒫 𝑋)
53, 4sylibr 225 . . . . . 6 (𝑔 ∈ (Fil‘𝑋) → 𝑔 ∈ 𝒫 𝒫 𝑋)
65a1d 25 . . . . 5 (𝑔 ∈ (Fil‘𝑋) → (𝐹𝑔𝑔 ∈ 𝒫 𝒫 𝑋))
72, 6mprgbir 3074 . . . 4 {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋
8 ssnum 9113 . . . 4 ((𝒫 𝒫 𝑋 ∈ dom card ∧ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ⊆ 𝒫 𝒫 𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
91, 7, 8sylancl 580 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∈ dom card)
10 ssid 3783 . . . . . . 7 𝐹𝐹
1110jctr 520 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
12 sseq2 3787 . . . . . . 7 (𝑔 = 𝐹 → (𝐹𝑔𝐹𝐹))
1312elrab 3519 . . . . . 6 (𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝐹 ∈ (Fil‘𝑋) ∧ 𝐹𝐹))
1411, 13sylibr 225 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
1514ne0d 4086 . . . 4 (𝐹 ∈ (Fil‘𝑋) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
1615adantr 472 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ≠ ∅)
17 simpr1 1248 . . . . . . . . . 10 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
18 ssrab 3840 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
1917, 18sylib 209 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → (𝑥 ⊆ (Fil‘𝑋) ∧ ∀𝑔𝑥 𝐹𝑔))
2019simpld 488 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ⊆ (Fil‘𝑋))
21 simpr2 1250 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ≠ ∅)
22 simpr3 1252 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → [] Or 𝑥)
23 sorpssun 7142 . . . . . . . . . 10 (( [] Or 𝑥 ∧ (𝑔𝑥𝑥)) → (𝑔) ∈ 𝑥)
2423ralrimivva 3118 . . . . . . . . 9 ( [] Or 𝑥 → ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥)
2522, 24syl 17 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥)
26 filuni 21968 . . . . . . . 8 ((𝑥 ⊆ (Fil‘𝑋) ∧ 𝑥 ≠ ∅ ∧ ∀𝑔𝑥𝑥 (𝑔) ∈ 𝑥) → 𝑥 ∈ (Fil‘𝑋))
2720, 21, 25, 26syl3anc 1490 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ (Fil‘𝑋))
28 n0 4095 . . . . . . . . 9 (𝑥 ≠ ∅ ↔ ∃ 𝑥)
29 ssel2 3756 . . . . . . . . . . . . . 14 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
30 sseq2 3787 . . . . . . . . . . . . . . 15 (𝑔 = → (𝐹𝑔𝐹))
3130elrab 3519 . . . . . . . . . . . . . 14 ( ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( ∈ (Fil‘𝑋) ∧ 𝐹))
3229, 31sylib 209 . . . . . . . . . . . . 13 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → ( ∈ (Fil‘𝑋) ∧ 𝐹))
3332simprd 489 . . . . . . . . . . . 12 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹)
34 ssuni 4618 . . . . . . . . . . . 12 ((𝐹𝑥) → 𝐹 𝑥)
3533, 34sylancom 582 . . . . . . . . . . 11 ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥) → 𝐹 𝑥)
3635ex 401 . . . . . . . . . 10 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥𝐹 𝑥))
3736exlimdv 2028 . . . . . . . . 9 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (∃ 𝑥𝐹 𝑥))
3828, 37syl5bi 233 . . . . . . . 8 (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} → (𝑥 ≠ ∅ → 𝐹 𝑥))
3917, 21, 38sylc 65 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝐹 𝑥)
40 sseq2 3787 . . . . . . . 8 (𝑔 = 𝑥 → (𝐹𝑔𝐹 𝑥))
4140elrab 3519 . . . . . . 7 ( 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ ( 𝑥 ∈ (Fil‘𝑋) ∧ 𝐹 𝑥))
4227, 39, 41sylanbrc 578 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥)) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔})
4342ex 401 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → ((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4443alrimiv 2022 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
4544adantr 472 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∀𝑥((𝑥 ⊆ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ 𝑥 ≠ ∅ ∧ [] Or 𝑥) → 𝑥 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}))
46 zornn0g 9580 . . 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)
479, 16, 45, 46syl3anc 1490 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝒫 𝒫 𝑋 ∈ dom card) → ∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓)
48 sseq2 3787 . . . . 5 (𝑔 = 𝑓 → (𝐹𝑔𝐹𝑓))
4948elrab 3519 . . . 4 (𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ↔ (𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓))
5030ralrab 3525 . . . 4 (∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 ↔ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓))
51 simpll 783 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
52 sstr2 3768 . . . . . . . . . . 11 (𝐹𝑓 → (𝑓𝐹))
5352imim1d 82 . . . . . . . . . 10 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓 → ¬ 𝑓)))
54 df-pss 3748 . . . . . . . . . . . . 13 (𝑓 ↔ (𝑓𝑓))
5554simplbi2 494 . . . . . . . . . . . 12 (𝑓 → (𝑓𝑓))
5655necon1bd 2955 . . . . . . . . . . 11 (𝑓 → (¬ 𝑓𝑓 = ))
5756a2i 14 . . . . . . . . . 10 ((𝑓 → ¬ 𝑓) → (𝑓𝑓 = ))
5853, 57syl6 35 . . . . . . . . 9 (𝐹𝑓 → ((𝐹 → ¬ 𝑓) → (𝑓𝑓 = )))
5958ralimdv 3110 . . . . . . . 8 (𝐹𝑓 → (∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6059imp 395 . . . . . . 7 ((𝐹𝑓 ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
6160adantll 705 . . . . . 6 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = ))
62 isufil2 21991 . . . . . 6 (𝑓 ∈ (UFil‘𝑋) ↔ (𝑓 ∈ (Fil‘𝑋) ∧ ∀ ∈ (Fil‘𝑋)(𝑓𝑓 = )))
6351, 61, 62sylanbrc 578 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝑓 ∈ (UFil‘𝑋))
64 simplr 785 . . . . 5 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → 𝐹𝑓)
6563, 64jca 507 . . . 4 (((𝑓 ∈ (Fil‘𝑋) ∧ 𝐹𝑓) ∧ ∀ ∈ (Fil‘𝑋)(𝐹 → ¬ 𝑓)) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6649, 50, 65syl2anb 591 . . 3 ((𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ∧ ∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓) → (𝑓 ∈ (UFil‘𝑋) ∧ 𝐹𝑓))
6766reximi2 3156 . 2 (∃𝑓 ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔}∀ ∈ {𝑔 ∈ (Fil‘𝑋) ∣ 𝐹𝑔} ¬ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)𝐹𝑓)
6847, 67syl 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 1107  wal 1650  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  cun 3730  wss 3732  wpss 3733  c0 4079  𝒫 cpw 4315   cuni 4594   Or wor 5197  dom cdm 5277  cfv 6068   [] crpss 7134  cardccrd 9012  Filcfil 21928  UFilcufil 21982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-rpss 7135  df-om 7264  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-card 9016  df-cda 9243  df-fbas 20016  df-fg 20017  df-fil 21929  df-ufil 21984
This theorem is referenced by:  filssufil  21995  numufl  21998
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