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Theorem csdfil 23902
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 4120 . . . . . 6 (𝑥 = 𝑦 → (𝑋𝑥) = (𝑋𝑦))
21breq1d 5153 . . . . 5 (𝑥 = 𝑦 → ((𝑋𝑥) ≺ 𝑋 ↔ (𝑋𝑦) ≺ 𝑋))
32elrab 3692 . . . 4 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
4 velpw 4605 . . . . 5 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
54anbi1i 624 . . . 4 ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋𝑦) ≺ 𝑋) ↔ (𝑦𝑋 ∧ (𝑋𝑦) ≺ 𝑋))
63, 5bitri 275 . . 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 simpl 482 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ dom card)
9 difid 4376 . . . 4 (𝑋𝑋) = ∅
10 infn0 9340 . . . . . 6 (ω ≼ 𝑋𝑋 ≠ ∅)
1110adantl 481 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12 0sdomg 9144 . . . . . 6 (𝑋 ∈ dom card → (∅ ≺ 𝑋𝑋 ≠ ∅))
1312adantr 480 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋𝑋 ≠ ∅))
1411, 13mpbird 257 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
159, 14eqbrtrid 5178 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋𝑋) ≺ 𝑋)
16 difeq2 4120 . . . . . 6 (𝑦 = 𝑋 → (𝑋𝑦) = (𝑋𝑋))
1716breq1d 5153 . . . . 5 (𝑦 = 𝑋 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1817sbcieg 3828 . . . 4 (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
1918adantr 480 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑋) ≺ 𝑋))
2015, 19mpbird 257 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋𝑦) ≺ 𝑋)
21 sdomirr 9154 . . 3 ¬ 𝑋𝑋
22 0ex 5307 . . . . 5 ∅ ∈ V
23 difeq2 4120 . . . . . . 7 (𝑦 = ∅ → (𝑋𝑦) = (𝑋 ∖ ∅))
24 dif0 4378 . . . . . . 7 (𝑋 ∖ ∅) = 𝑋
2523, 24eqtrdi 2793 . . . . . 6 (𝑦 = ∅ → (𝑋𝑦) = 𝑋)
2625breq1d 5153 . . . . 5 (𝑦 = ∅ → ((𝑋𝑦) ≺ 𝑋𝑋𝑋))
2722, 26sbcie 3830 . . . 4 ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋)
2827a1i 11 . . 3 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([∅ / 𝑦](𝑋𝑦) ≺ 𝑋𝑋𝑋))
2921, 28mtbiri 327 . 2 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋𝑦) ≺ 𝑋)
30 simp1l 1198 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → 𝑋 ∈ dom card)
3130difexd 5331 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑤) ∈ V)
32 sscon 4143 . . . . . 6 (𝑤𝑧 → (𝑋𝑧) ⊆ (𝑋𝑤))
33323ad2ant3 1136 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ⊆ (𝑋𝑤))
34 ssdomg 9040 . . . . 5 ((𝑋𝑤) ∈ V → ((𝑋𝑧) ⊆ (𝑋𝑤) → (𝑋𝑧) ≼ (𝑋𝑤)))
3531, 33, 34sylc 65 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → (𝑋𝑧) ≼ (𝑋𝑤))
36 domsdomtr 9152 . . . . 5 (((𝑋𝑧) ≼ (𝑋𝑤) ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋𝑧) ≺ 𝑋)
3736ex 412 . . . 4 ((𝑋𝑧) ≼ (𝑋𝑤) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
3835, 37syl 17 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ((𝑋𝑤) ≺ 𝑋 → (𝑋𝑧) ≺ 𝑋))
39 vex 3484 . . . 4 𝑤 ∈ V
40 difeq2 4120 . . . . 5 (𝑦 = 𝑤 → (𝑋𝑦) = (𝑋𝑤))
4140breq1d 5153 . . . 4 (𝑦 = 𝑤 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋))
4239, 41sbcie 3830 . . 3 ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑤) ≺ 𝑋)
43 vex 3484 . . . 4 𝑧 ∈ V
44 difeq2 4120 . . . . 5 (𝑦 = 𝑧 → (𝑋𝑦) = (𝑋𝑧))
4544breq1d 5153 . . . 4 (𝑦 = 𝑧 → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋))
4643, 45sbcie 3830 . . 3 ([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋𝑧) ≺ 𝑋)
4738, 42, 463imtr4g 296 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑧) → ([𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋))
48 infunsdom 10253 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋)) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
4948ex 412 . . . . 5 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋))
50 difindi 4292 . . . . . 6 (𝑋 ∖ (𝑧𝑤)) = ((𝑋𝑧) ∪ (𝑋𝑤))
5150breq1i 5150 . . . . 5 ((𝑋 ∖ (𝑧𝑤)) ≺ 𝑋 ↔ ((𝑋𝑧) ∪ (𝑋𝑤)) ≺ 𝑋)
5249, 51imbitrrdi 252 . . . 4 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
53523ad2ant1 1134 . . 3 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5446, 42anbi12i 628 . . 3 (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) ↔ ((𝑋𝑧) ≺ 𝑋 ∧ (𝑋𝑤) ≺ 𝑋))
5543inex1 5317 . . . 4 (𝑧𝑤) ∈ V
56 difeq2 4120 . . . . 5 (𝑦 = (𝑧𝑤) → (𝑋𝑦) = (𝑋 ∖ (𝑧𝑤)))
5756breq1d 5153 . . . 4 (𝑦 = (𝑧𝑤) → ((𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋))
5855, 57sbcie 3830 . . 3 ([(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧𝑤)) ≺ 𝑋)
5953, 54, 583imtr4g 296 . 2 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧𝑋𝑤𝑋) → (([𝑧 / 𝑦](𝑋𝑦) ≺ 𝑋[𝑤 / 𝑦](𝑋𝑦) ≺ 𝑋) → [(𝑧𝑤) / 𝑦](𝑋𝑦) ≺ 𝑋))
607, 8, 20, 29, 47, 59isfild 23866 1 ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {crab 3436  Vcvv 3480  [wsbc 3788  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  cfv 6561  ωcom 7887  cdom 8983  csdm 8984  cardccrd 9975  Filcfil 23853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-dju 9941  df-card 9979  df-fbas 21361  df-fil 23854
This theorem is referenced by:  ufilen  23938
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