Theorem csdfil 23842

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 4073	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑋 ∖ 𝑥) = (𝑋 ∖ 𝑦))
2	1	breq1d 5109	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑋 ∖ 𝑥) ≺ 𝑋 ↔ (𝑋 ∖ 𝑦) ≺ 𝑋))
3	2	elrab 3647	. . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
4		velpw 4560	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
5	4	anbi1i 625	. . . 4 ⊢ ((𝑦 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋) ↔ (𝑦 ⊆ 𝑋 ∧ (𝑋 ∖ 𝑦) ≺ 𝑋))
6	3, 5	bitri 275	. . 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 4329	. . . 4 ⊢ (𝑋 ∖ 𝑋) = ∅
10		infn0 9206	. . . . . 6 ⊢ (ω ≼ 𝑋 → 𝑋 ≠ ∅)
11	10	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → 𝑋 ≠ ∅)
12		0sdomg 9038	. . . . . 6 ⊢ (𝑋 ∈ dom card → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
13	12	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (∅ ≺ 𝑋 ↔ 𝑋 ≠ ∅))
14	11, 13	mpbird 257	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ∅ ≺ 𝑋)
15	9, 14	eqbrtrid 5134	. . 3 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (𝑋 ∖ 𝑋) ≺ 𝑋)
16		difeq2 4073	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
17	16	breq1d 5109	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑋) ≺ 𝑋))
18	17	sbcieg 3781	. . . 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 257	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → [𝑋 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
21		sdomirr 9046	. . 3 ⊢ ¬ 𝑋 ≺ 𝑋
22		0ex 5253	. . . . 5 ⊢ ∅ ∈ V
23		difeq2 4073	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
24		dif0 4331	. . . . . . 7 ⊢ (𝑋 ∖ ∅) = 𝑋
25	23, 24	eqtrdi 2788	. . . . . 6 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = 𝑋)
26	25	breq1d 5109	. . . . 5 ⊢ (𝑦 = ∅ → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ 𝑋 ≺ 𝑋))
27	22, 26	sbcie 3783	. . . 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 327	. 2 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → ¬ [∅ / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋)
30		simp1l 1199	. . . . . 6 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → 𝑋 ∈ dom card)
31	30	difexd 5277	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑤) ∈ V)
32		sscon 4096	. . . . . 6 ⊢ (𝑤 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
33	32	3ad2ant3 1136	. . . . 5 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑤))
34		ssdomg 8941	. . . . 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 9044	. . . . 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 3445	. . . 4 ⊢ 𝑤 ∈ V
40		difeq2 4073	. . . . 5 ⊢ (𝑦 = 𝑤 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑤))
41	40	breq1d 5109	. . . 4 ⊢ (𝑦 = 𝑤 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋))
42	39, 41	sbcie 3783	. . 3 ⊢ ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑤) ≺ 𝑋)
43		vex 3445	. . . 4 ⊢ 𝑧 ∈ V
44		difeq2 4073	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑧))
45	44	breq1d 5109	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋))
46	43, 45	sbcie 3783	. . 3 ⊢ ([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ 𝑧) ≺ 𝑋)
47	38, 42, 46	3imtr4g 296	. 2 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑧) → ([𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 → [𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋))
48		infunsdom 10127	. . . . . 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 4245	. . . . . 6 ⊢ (𝑋 ∖ (𝑧 ∩ 𝑤)) = ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤))
51	50	breq1i 5106	. . . . 5 ⊢ ((𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋 ↔ ((𝑋 ∖ 𝑧) ∪ (𝑋 ∖ 𝑤)) ≺ 𝑋)
52	49, 51	imbitrrdi 252	. . . 4 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
53	52	3ad2ant1 1134	. . 3 ⊢ (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝑧 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋) → (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
54	46, 42	anbi12i 629	. . 3 ⊢ (([𝑧 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ∧ [𝑤 / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋) ↔ ((𝑋 ∖ 𝑧) ≺ 𝑋 ∧ (𝑋 ∖ 𝑤) ≺ 𝑋))
55	43	inex1 5263	. . . 4 ⊢ (𝑧 ∩ 𝑤) ∈ V
56		difeq2 4073	. . . . 5 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → (𝑋 ∖ 𝑦) = (𝑋 ∖ (𝑧 ∩ 𝑤)))
57	56	breq1d 5109	. . . 4 ⊢ (𝑦 = (𝑧 ∩ 𝑤) → ((𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋))
58	55, 57	sbcie 3783	. . 3 ⊢ ([(𝑧 ∩ 𝑤) / 𝑦](𝑋 ∖ 𝑦) ≺ 𝑋 ↔ (𝑋 ∖ (𝑧 ∩ 𝑤)) ≺ 𝑋)
59	53, 54, 58	3imtr4g 296	. 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 23806	1 ⊢ ((𝑋 ∈ dom card ∧ ω ≼ 𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ≺ 𝑋} ∈ (Fil‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  {crab 3400  Vcvv 3441  [wsbc 3741   ∖ cdif 3899   ∪ cun 3900   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286  𝒫 cpw 4555   class class class wbr 5099  dom cdm 5625  ‘cfv 6493  ωcom 7810   ≼ cdom 8885   ≺ csdm 8886  cardccrd 9851  Filcfil 23793

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-dju 9817  df-card 9855  df-fbas 21310  df-fil 23794

This theorem is referenced by:  ufilen  23878