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Theorem csdfil 21508
 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 3684 . . . . . 6 (𝑥 = 𝑦 → (𝑋𝑥) = (𝑋𝑦))
21breq1d 4593 . . . . 5 (𝑥 = 𝑦 → ((𝑋𝑥) ≺ 𝑋 ↔ (𝑋𝑦) ≺ 𝑋))
32elrab 3331 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
4 selpw 4115 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 727 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋) ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
63, 5bitri 263 . . 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 3185 . . 3 (𝑋 ∈ dom card → 𝑋 ∈ V)
98adantr 480 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ V)
10 difid 3902 . . . 4 (𝑋𝑋) = ∅
11 infn0 8107 . . . . . 6 (ω ≼ 𝑋𝑋 ≠ ∅)
1211adantl 481 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
13 0sdomg 7974 . . . . . 6 (𝑋 ∈ dom card → (∅ ≺ 𝑋𝑋 ≠ ∅))
1413adantr 480 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋𝑋 ≠ ∅))
1512, 14mpbird 246 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
1610, 15syl5eqbr 4618 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋𝑋) ≺ 𝑋)
17 difeq2 3684 . . . . . 6 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
1817breq1d 4593 . . . . 5 (𝑦 = 𝑋 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1918sbcieg 3435 . . . 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 246 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋)
22 sdomirr 7982 . . 3 ¬ 𝑋𝑋
23 0ex 4718 . . . . 5 ∅ ∈ V
24 difeq2 3684 . . . . . . 7 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
25 dif0 3904 . . . . . . 7 (𝑋 ∖ ∅) = 𝑋
2624, 25syl6eq 2660 . . . . . 6 (𝑦 = ∅ → (𝑋𝑦) = 𝑋)
2726breq1d 4593 . . . . 5 (𝑦 = ∅ → ((𝑋𝑦) ≺ 𝑋𝑋𝑋))
2823, 27sbcie 3437 . . . 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 1078 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → 𝑋 ∈ dom card)
32 difexg 4735 . . . . . 6 (𝑋 ∈ dom card → (𝑋𝑤) ∈ V)
3331, 32syl 17 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑤) ∈ V)
34 sscon 3706 . . . . . 6 (𝑤𝑧 → (𝑋𝑧) ⊆ (𝑋𝑤))
35343ad2ant3 1077 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ⊆ (𝑋𝑤))
36 ssdomg 7887 . . . . 5 ((𝑋𝑤) ∈ V → ((𝑋𝑧) ⊆ (𝑋𝑤) → (𝑋𝑧) ≼ (𝑋𝑤)))
3733, 35, 36sylc 63 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ≼ (𝑋𝑤))
38 domsdomtr 7980 . . . . 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 3176 . . . 4 𝑤 ∈ V
42 difeq2 3684 . . . . 5 (𝑦 = 𝑤 → (𝑋𝑦) = (𝑋𝑤))
4342breq1d 4593 . . . 4 (𝑦 = 𝑤 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋))
4441, 43sbcie 3437 . . 3 ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋)
45 vex 3176 . . . 4 𝑧 ∈ V
46 difeq2 3684 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦) = (𝑋𝑧))
4746breq1d 4593 . . . 4 (𝑦 = 𝑧 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋))
4845, 47sbcie 3437 . . 3 ([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋)
4940, 44, 483imtr4g 284 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋))
50 infunsdom 8919 . . . . . 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 3840 . . . . . 6 (𝑋 ∖ (𝑧𝑤)) = ((𝑋𝑧) ∪ (𝑋𝑤))
5352breq1i 4590 . . . . 5 ((𝑋 ∖ (𝑧𝑤)) ≺ 𝑋 ↔ ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
5451, 53syl6ibr 241 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
55543ad2ant1 1075 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5648, 44anbi12i 729 . . 3 (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) ↔ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋))
5745inex1 4727 . . . 4 (𝑧𝑤) ∈ V
58 difeq2 3684 . . . . 5 (𝑦 = (𝑧𝑤) → (𝑋𝑦) = (𝑋 ∖ (𝑧𝑤)))
5958breq1d 4593 . . . 4 (𝑦 = (𝑧𝑤) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
6057, 59sbcie 3437 . . 3 ([(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋)
6155, 56, 603imtr4g 284 . 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 21472 1 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900  Vcvv 3173  [wsbc 3402   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  ωcom 6957   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644  Filcfil 21459 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873  df-fbas 19564  df-fil 21460 This theorem is referenced by:  ufilen  21544
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