Theorem csdfil 23618

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 ∣ (𝑋 ∖ 𝑥) ≺ 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.

Step	Hyp	Ref	Expression
1		difeq2 4115	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑦))
2	1	breq1d 5157	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑋 ∖ 𝑥) ≺ 𝑋 ↔ (𝑋 ∖ 𝑦) ≺ 𝑋))
3	2	elrab 3682	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
4		velpw 4606	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 622	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋) ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
6	3, 5	bitri 274	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
7	6	a1i 11	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋)))
8		simpl 481	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ dom card)
9		difid 4369	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
10		infn0 9309	. . . . . 6 ⊢ (ω ≼ 𝑋 → 𝑋 ≠ ∅)
11	10	adantl 480	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12		0sdomg 9106	. . . . . 6 ⊢ (𝑋 ∈ dom card → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
13	12	adantr 479	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 256	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
15	9, 14	eqbrtrid 5182	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋 ∖ 𝑋) ≺ 𝑋)
16		difeq2 4115	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
17	16	breq1d 5157	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
18	17	sbcieg 3816	. . . 4 ⊢ (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
19	18	adantr 479	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
20	15, 19	mpbird 256	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
21		sdomirr 9116	. . 3 ⊢ ¬ 𝑋 ≺ 𝑋
22		0ex 5306	. . . . 5 ⊢ ∅ ∈ V
23		difeq2 4115	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
24		dif0 4371	. . . . . . 7 ⊢ (𝑋 ∖ ∅) = 𝑋
25	23, 24	eqtrdi 2786	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = 𝑋)
26	25	breq1d 5157	. . . . 5 ⊢ (𝑦 = ∅ → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
27	22, 26	sbcie 3819	. . . 4 ⊢ ([∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋)
28	27	a1i 11	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
29	21, 28	mtbiri 326	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
30		simp1l 1195	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → 𝑋 ∈ dom card)
31	30	difexd 5328	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑤) ∈ V)
32		sscon 4137	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
33	32	3ad2ant3 1133	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
34		ssdomg 8998	. . . . 5 ⊢ ((𝑋 ∖ 𝑤) ∈ V → ((𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤)))
35	31, 33, 34	sylc 65	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤))
36		domsdomtr 9114	. . . . 5 ⊢ (((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ 𝑧) ≺ 𝑋)
37	36	ex 411	. . . 4 ⊢ ((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
38	35, 37	syl 17	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
39		vex 3476	. . . 4 ⊢ 𝑤 ∈ V
40		difeq2 4115	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑤))
41	40	breq1d 5157	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋))
42	39, 41	sbcie 3819	. . 3 ⊢ ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋)
43		vex 3476	. . . 4 ⊢ 𝑧 ∈ V
44		difeq2 4115	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑧))
45	44	breq1d 5157	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋))
46	43, 45	sbcie 3819	. . 3 ⊢ ([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋)
47	38, 42, 46	3imtr4g 295	. 2 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 → [𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋))
48		infunsdom 10211	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋)) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
49	48	ex 411	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋))
50		difindi 4280	. . . . . 6 ⊢ (𝑋 ∖ (𝑧 ∩ 𝑤)) = ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤))
51	50	breq1i 5154	. . . . 5 ⊢ ((𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋 ↔ ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
52	49, 51	imbitrrdi 251	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
53	52	3ad2ant1 1131	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
54	46, 42	anbi12i 625	. . 3 ⊢ (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) ↔ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋))
55	43	inex1 5316	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
56		difeq2 4115	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝑋 ∖ 𝑦) = (𝑋 ∖ (𝑧 ∩ 𝑤)))
57	56	breq1d 5157	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
58	55, 57	sbcie 3819	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋)
59	53, 54, 58	3imtr4g 295	. 2 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) → [(𝑧 ∩ 𝑤) / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋))
60	7, 8, 20, 29, 47, 59	isfild 23582	1 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  {crab 3430  Vcvv 3472  [wsbc 3776   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601   class class class wbr 5147  dom cdm 5675  ‘cfv 6542  ωcom 7857   ≼ cdom 8939   ≺ csdm 8940  cardccrd 9932  Filcfil 23569

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-dju 9898  df-card 9936  df-fbas 21141  df-fil 23570

This theorem is referenced by:  ufilen  23654