Theorem csdfil 22953

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 4047	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑦))
2	1	breq1d 5080	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑋 ∖ 𝑥) ≺ 𝑋 ↔ (𝑋 ∖ 𝑦) ≺ 𝑋))
3	2	elrab 3617	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
4		velpw 4535	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 623	. . . 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 482	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ dom card)
9		difid 4301	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
10		infn0 9006	. . . . . 6 ⊢ (ω ≼ 𝑋 → 𝑋 ≠ ∅)
11	10	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12		0sdomg 8842	. . . . . 6 ⊢ (𝑋 ∈ dom card → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
13	12	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 256	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
15	9, 14	eqbrtrid 5105	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋 ∖ 𝑋) ≺ 𝑋)
16		difeq2 4047	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
17	16	breq1d 5080	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
18	17	sbcieg 3751	. . . 4 ⊢ (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
19	18	adantr 480	. . 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 8850	. . 3 ⊢ ¬ 𝑋 ≺ 𝑋
22		0ex 5226	. . . . 5 ⊢ ∅ ∈ V
23		difeq2 4047	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
24		dif0 4303	. . . . . . 7 ⊢ (𝑋 ∖ ∅) = 𝑋
25	23, 24	eqtrdi 2795	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = 𝑋)
26	25	breq1d 5080	. . . . 5 ⊢ (𝑦 = ∅ → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
27	22, 26	sbcie 3754	. . . 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 5248	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑤) ∈ V)
32		sscon 4069	. . . . . 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 8741	. . . . 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 8848	. . . . 5 ⊢ (((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ 𝑧) ≺ 𝑋)
37	36	ex 412	. . . 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 3426	. . . 4 ⊢ 𝑤 ∈ V
40		difeq2 4047	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑤))
41	40	breq1d 5080	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋))
42	39, 41	sbcie 3754	. . 3 ⊢ ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋)
43		vex 3426	. . . 4 ⊢ 𝑧 ∈ V
44		difeq2 4047	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑧))
45	44	breq1d 5080	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋))
46	43, 45	sbcie 3754	. . 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 9901	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋)) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
49	48	ex 412	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋))
50		difindi 4212	. . . . . 6 ⊢ (𝑋 ∖ (𝑧 ∩ 𝑤)) = ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤))
51	50	breq1i 5077	. . . . 5 ⊢ ((𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋 ↔ ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
52	49, 51	syl6ibr 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 626	. . 3 ⊢ (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) ↔ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋))
55	43	inex1 5236	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
56		difeq2 4047	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝑋 ∖ 𝑦) = (𝑋 ∖ (𝑧 ∩ 𝑤)))
57	56	breq1d 5080	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
58	55, 57	sbcie 3754	. . 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 22917	1 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  {crab 3067  Vcvv 3422  [wsbc 3711   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  ωcom 7687   ≼ cdom 8689   ≺ csdm 8690  cardccrd 9624  Filcfil 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-dju 9590  df-card 9628  df-fbas 20507  df-fil 22905

This theorem is referenced by:  ufilen  22989