Theorem csdfil 22499

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 4044	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑦))
2	1	breq1d 5040	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑋 ∖ 𝑥) ≺ 𝑋 ↔ (𝑋 ∖ 𝑦) ≺ 𝑋))
3	2	elrab 3628	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
4		velpw 4502	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 626	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋) ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
6	3, 5	bitri 278	. . 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 486	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ∈ dom card)
9		difid 4284	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
10		infn0 8764	. . . . . 6 ⊢ (ω ≼ 𝑋 → 𝑋 ≠ ∅)
11	10	adantl 485	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12		0sdomg 8630	. . . . . 6 ⊢ (𝑋 ∈ dom card → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
13	12	adantr 484	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 260	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
15	9, 14	eqbrtrid 5065	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋 ∖ 𝑋) ≺ 𝑋)
16		difeq2 4044	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
17	16	breq1d 5040	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
18	17	sbcieg 3758	. . . 4 ⊢ (𝑋 ∈ dom card → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
19	18	adantr 484	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ([𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
20	15, 19	mpbird 260	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
21		sdomirr 8638	. . 3 ⊢ ¬ 𝑋 ≺ 𝑋
22		0ex 5175	. . . . 5 ⊢ ∅ ∈ V
23		difeq2 4044	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
24		dif0 4286	. . . . . . 7 ⊢ (𝑋 ∖ ∅) = 𝑋
25	23, 24	eqtrdi 2849	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = 𝑋)
26	25	breq1d 5040	. . . . 5 ⊢ (𝑦 = ∅ → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
27	22, 26	sbcie 3760	. . . 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 330	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
30		simp1l 1194	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → 𝑋 ∈ dom card)
31		difexg 5195	. . . . . 6 ⊢ (𝑋 ∈ dom card → (𝑋 ∖ 𝑤) ∈ V)
32	30, 31	syl 17	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑤) ∈ V)
33		sscon 4066	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
34	33	3ad2ant3 1132	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
35		ssdomg 8538	. . . . 5 ⊢ ((𝑋 ∖ 𝑤) ∈ V → ((𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤)))
36	32, 34, 35	sylc 65	. . . 4 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤))
37		domsdomtr 8636	. . . . 5 ⊢ (((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ 𝑧) ≺ 𝑋)
38	37	ex 416	. . . 4 ⊢ ((𝑋 ∖ 𝑧) ≼ (𝑋 ∖ 𝑤) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
39	36, 38	syl 17	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ((𝑋 ∖ 𝑤) ≺ 𝑋 → (𝑋 ∖ 𝑧) ≺ 𝑋))
40		vex 3444	. . . 4 ⊢ 𝑤 ∈ V
41		difeq2 4044	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑤))
42	41	breq1d 5040	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋))
43	40, 42	sbcie 3760	. . 3 ⊢ ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋)
44		vex 3444	. . . 4 ⊢ 𝑧 ∈ V
45		difeq2 4044	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑧))
46	45	breq1d 5040	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋))
47	44, 46	sbcie 3760	. . 3 ⊢ ([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋)
48	39, 43, 47	3imtr4g 299	. 2 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 → [𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋))
49		infunsdom 9625	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋)) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
50	49	ex 416	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋))
51		difindi 4208	. . . . . 6 ⊢ (𝑋 ∖ (𝑧 ∩ 𝑤)) = ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤))
52	51	breq1i 5037	. . . . 5 ⊢ ((𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋 ↔ ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
53	50, 52	syl6ibr 255	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
54	53	3ad2ant1 1130	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
55	47, 43	anbi12i 629	. . 3 ⊢ (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) ↔ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋))
56	44	inex1 5185	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
57		difeq2 4044	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝑋 ∖ 𝑦) = (𝑋 ∖ (𝑧 ∩ 𝑤)))
58	57	breq1d 5040	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
59	56, 58	sbcie 3760	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋)
60	54, 55, 59	3imtr4g 299	. 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))
61	7, 8, 20, 29, 48, 60	isfild 22463	1 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441  [wsbc 3720   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497   class class class wbr 5030  dom cdm 5519  ‘cfv 6324  ωcom 7560   ≼ cdom 8490   ≺ csdm 8491  cardccrd 9348  Filcfil 22450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-dju 9314  df-card 9352  df-fbas 20088  df-fil 22451

This theorem is referenced by:  ufilen  22535