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Theorem csdfil 21745
Description: The set of all elements whose complement is dominated by the base set is a filter. (Contributed by Mario Carneiro, 14-Dec-2013.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
csdfil ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Distinct variable group:   𝑥,𝑋

Proof of Theorem csdfil
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difeq2 3755 . . . . . 6 (𝑥 = 𝑦 → (𝑋𝑥) = (𝑋𝑦))
21breq1d 4695 . . . . 5 (𝑥 = 𝑦 → ((𝑋𝑥) ≺ 𝑋 ↔ (𝑋𝑦) ≺ 𝑋))
32elrab 3396 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
4 selpw 4198 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 731 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋) ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
63, 5bitri 264 . . 3 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
76a1i 11 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋)))
8 elex 3243 . . 3 (𝑋 ∈ dom card → 𝑋 ∈ V)
98adantr 480 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ V)
10 difid 3981 . . . 4 (𝑋𝑋) = ∅
11 infn0 8263 . . . . . 6 (ω ≼ 𝑋𝑋 ≠ ∅)
1211adantl 481 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
13 0sdomg 8130 . . . . . 6 (𝑋 ∈ dom card → (∅ ≺ 𝑋𝑋 ≠ ∅))
1413adantr 480 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋𝑋 ≠ ∅))
1512, 14mpbird 247 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
1610, 15syl5eqbr 4720 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋𝑋) ≺ 𝑋)
17 difeq2 3755 . . . . . 6 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
1817breq1d 4695 . . . . 5 (𝑦 = 𝑋 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1918sbcieg 3501 . . . 4 (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
2019adantr 480 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
2116, 20mpbird 247 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋)
22 sdomirr 8138 . . 3 ¬ 𝑋𝑋
23 0ex 4823 . . . . 5 ∅ ∈ V
24 difeq2 3755 . . . . . . 7 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
25 dif0 3983 . . . . . . 7 (𝑋 ∖ ∅) = 𝑋
2624, 25syl6eq 2701 . . . . . 6 (𝑦 = ∅ → (𝑋𝑦) = 𝑋)
2726breq1d 4695 . . . . 5 (𝑦 = ∅ → ((𝑋𝑦) ≺ 𝑋𝑋𝑋))
2823, 27sbcie 3503 . . . 4 ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋)
2928a1i 11 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋))
3022, 29mtbiri 316 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋𝑦) ≺ 𝑋)
31 simp1l 1105 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → 𝑋 ∈ dom card)
32 difexg 4841 . . . . . 6 (𝑋 ∈ dom card → (𝑋𝑤) ∈ V)
3331, 32syl 17 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑤) ∈ V)
34 sscon 3777 . . . . . 6 (𝑤𝑧 → (𝑋𝑧) ⊆ (𝑋𝑤))
35343ad2ant3 1104 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ⊆ (𝑋𝑤))
36 ssdomg 8043 . . . . 5 ((𝑋𝑤) ∈ V → ((𝑋𝑧) ⊆ (𝑋𝑤) → (𝑋𝑧) ≼ (𝑋𝑤)))
3733, 35, 36sylc 65 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ≼ (𝑋𝑤))
38 domsdomtr 8136 . . . . 5 (((𝑋𝑧) ≼ (𝑋𝑤) ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋𝑧) ≺ 𝑋)
3938ex 449 . . . 4 ((𝑋𝑧) ≼ (𝑋𝑤) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
4037, 39syl 17 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
41 vex 3234 . . . 4 𝑤 ∈ V
42 difeq2 3755 . . . . 5 (𝑦 = 𝑤 → (𝑋𝑦) = (𝑋𝑤))
4342breq1d 4695 . . . 4 (𝑦 = 𝑤 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋))
4441, 43sbcie 3503 . . 3 ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋)
45 vex 3234 . . . 4 𝑧 ∈ V
46 difeq2 3755 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦) = (𝑋𝑧))
4746breq1d 4695 . . . 4 (𝑦 = 𝑧 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋))
4845, 47sbcie 3503 . . 3 ([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋)
4940, 44, 483imtr4g 285 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋))
50 infunsdom 9074 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋)) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
5150ex 449 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋))
52 difindi 3914 . . . . . 6 (𝑋 ∖ (𝑧𝑤)) = ((𝑋𝑧) ∪ (𝑋𝑤))
5352breq1i 4692 . . . . 5 ((𝑋 ∖ (𝑧𝑤)) ≺ 𝑋 ↔ ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
5451, 53syl6ibr 242 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
55543ad2ant1 1102 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5648, 44anbi12i 733 . . 3 (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) ↔ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋))
5745inex1 4832 . . . 4 (𝑧𝑤) ∈ V
58 difeq2 3755 . . . . 5 (𝑦 = (𝑧𝑤) → (𝑋𝑦) = (𝑋 ∖ (𝑧𝑤)))
5958breq1d 4695 . . . 4 (𝑦 = (𝑧𝑤) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
6057, 59sbcie 3503 . . 3 ([(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋)
6155, 56, 603imtr4g 285 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) → [(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋))
627, 9, 21, 30, 49, 61isfild 21709 1 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  [wsbc 3468  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191   class class class wbr 4685  dom cdm 5143  cfv 5926  ωcom 7107  cdom 7995  csdm 7996  cardccrd 8799  Filcfil 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-oi 8456  df-card 8803  df-cda 9028  df-fbas 19791  df-fil 21697
This theorem is referenced by:  ufilen  21781
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