Theorem csdfil 21698

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 3722	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑦))
2	1	breq1d 4663	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑋 ∖ 𝑥) ≺ 𝑋 ↔ (𝑋 ∖ 𝑦) ≺ 𝑋))
3	2	elrab 3363	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
4		selpw 4165	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 731	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋) ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
6	3, 5	bitri 264	. . 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		elex 3212	. . 3 ⊢ (𝑋 ∈ dom card → 𝑋 ∈ V)
9	8	adantr 481	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ V)
10		difid 3948	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
11		infn0 8222	. . . . . 6 ⊢ (ω ≼ 𝑋 → 𝑋 ≠ ∅)
12	11	adantl 482	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
13		0sdomg 8089	. . . . . 6 ⊢ (𝑋 ∈ dom card → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
14	13	adantr 481	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
15	12, 14	mpbird 247	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
16	10, 15	syl5eqbr 4688	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋 ∖ 𝑋) ≺ 𝑋)
17		difeq2 3722	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
18	17	breq1d 4663	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
19	18	sbcieg 3468	. . . 4 ⊢ (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
20	19	adantr 481	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
21	16, 20	mpbird 247	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
22		sdomirr 8097	. . 3 ⊢ ¬ 𝑋 ≺ 𝑋
23		0ex 4790	. . . . 5 ⊢ ∅ ∈ V
24		difeq2 3722	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
25		dif0 3950	. . . . . . 7 ⊢ (𝑋 ∖ ∅) = 𝑋
26	24, 25	syl6eq 2672	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = 𝑋)
27	26	breq1d 4663	. . . . 5 ⊢ (𝑦 = ∅ → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
28	23, 27	sbcie 3470	. . . 4 ⊢ ([∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋)
29	28	a1i 11	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
30	22, 29	mtbiri 317	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
31		simp1l 1085	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → 𝑋 ∈ dom card)
32		difexg 4808	. . . . . 6 ⊢ (𝑋 ∈ dom card → (𝑋 ∖ 𝑤) ∈ V)
33	31, 32	syl 17	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑤) ∈ V)
34		sscon 3744	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
35	34	3ad2ant3 1084	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
36		ssdomg 8001	. . . . 5 ⊢ ((𝑋 ∖ 𝑤) ∈ V → ((𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤)))
37	33, 35, 36	sylc 65	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤))
38		domsdomtr 8095	. . . . 5 ⊢ (((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ 𝑧) ≺ 𝑋)
39	38	ex 450	. . . 4 ⊢ ((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
40	37, 39	syl 17	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
41		vex 3203	. . . 4 ⊢ 𝑤 ∈ V
42		difeq2 3722	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑤))
43	42	breq1d 4663	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋))
44	41, 43	sbcie 3470	. . 3 ⊢ ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋)
45		vex 3203	. . . 4 ⊢ 𝑧 ∈ V
46		difeq2 3722	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑧))
47	46	breq1d 4663	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋))
48	45, 47	sbcie 3470	. . 3 ⊢ ([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋)
49	40, 44, 48	3imtr4g 285	. 2 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 → [𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋))
50		infunsdom 9036	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋)) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
51	50	ex 450	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋))
52		difindi 3881	. . . . . 6 ⊢ (𝑋 ∖ (𝑧 ∩ 𝑤)) = ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤))
53	52	breq1i 4660	. . . . 5 ⊢ ((𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋 ↔ ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
54	51, 53	syl6ibr 242	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
55	54	3ad2ant1 1082	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
56	48, 44	anbi12i 733	. . 3 ⊢ (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) ↔ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋))
57	45	inex1 4799	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
58		difeq2 3722	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝑋 ∖ 𝑦) = (𝑋 ∖ (𝑧 ∩ 𝑤)))
59	58	breq1d 4663	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
60	57, 59	sbcie 3470	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋)
61	55, 56, 60	3imtr4g 285	. 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))
62	7, 9, 21, 30, 49, 61	isfild 21662	1 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  Vcvv 3200  [wsbc 3435   ∖ cdif 3571   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  ωcom 7065   ≼ cdom 7953   ≺ csdm 7954  cardccrd 8761  Filcfil 21649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-cda 8990  df-fbas 19743  df-fil 21650

This theorem is referenced by:  ufilen  21734