Theorem ufinffr 22531

Description: An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.)

Assertion

Ref	Expression
ufinffr	⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅))

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝑓,𝑋

Proof of Theorem ufinffr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ominf 8724	. . . . 5 ⊢ ¬ ω ∈ Fin
2		domfi 8733	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ ω ≼ 𝐴) → ω ∈ Fin)
3	2	expcom 416	. . . . 5 ⊢ (ω ≼ 𝐴 → (𝐴 ∈ Fin → ω ∈ Fin))
4	1, 3	mtoi 201	. . . 4 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ∈ Fin)
5		cfinfil 22495	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))
6	4, 5	syl3an3 1161	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋))
7		filssufil 22514	. . 3 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ∈ (Fil‘𝑋) → ∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓)
8	6, 7	syl 17	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓)
9		difeq2 4092	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = (𝐴 ∖ 𝐴))
10		difid 4329	. . . . . . . 8 ⊢ (𝐴 ∖ 𝐴) = ∅
11	9, 10	syl6eq 2872	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐴 ∖ 𝑥) = ∅)
12	11	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐴 ∖ 𝑥) ∈ Fin ↔ ∅ ∈ Fin))
13		elpw2g 5239	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
14	13	biimpar 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋) → 𝐴 ∈ 𝒫 𝑋)
15	14	3adant3 1128	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → 𝐴 ∈ 𝒫 𝑋)
16		0fin 8740	. . . . . . 7 ⊢ ∅ ∈ Fin
17	16	a1i 11	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∅ ∈ Fin)
18	12, 15, 17	elrabd 3681	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin})
19		ssel 3960	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → (𝐴 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} → 𝐴 ∈ 𝑓))
20	18, 19	syl5com 31	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → 𝐴 ∈ 𝑓))
21		intss 4889	. . . . . 6 ⊢ ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → ∩ 𝑓 ⊆ ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin})
22		neldifsn 4718	. . . . . . . . . 10 ⊢ ¬ 𝑦 ∈ (𝐴 ∖ {𝑦})
23		elinti 4877	. . . . . . . . . 10 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} → ((𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} → 𝑦 ∈ (𝐴 ∖ {𝑦})))
24	22, 23	mtoi 201	. . . . . . . . 9 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} → ¬ (𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin})
25		difeq2 4092	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → (𝐴 ∖ 𝑥) = (𝐴 ∖ (𝐴 ∖ {𝑦})))
26	25	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 ∖ {𝑦}) → ((𝐴 ∖ 𝑥) ∈ Fin ↔ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin))
27		simp2 1133	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → 𝐴 ⊆ 𝑋)
28	27	ssdifssd 4118	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ⊆ 𝑋)
29		elpw2g 5239	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐵 → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝑋))
30	29	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ((𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋 ↔ (𝐴 ∖ {𝑦}) ⊆ 𝑋))
31	28, 30	mpbird 259	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ∈ 𝒫 𝑋)
32		snfi 8588	. . . . . . . . . . . 12 ⊢ {𝑦} ∈ Fin
33		eldif 3945	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∖ {𝑦})))
34		eldif 3945	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
35	34	notbii 322	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
36		iman 404	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑦}) ↔ ¬ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ {𝑦}))
37	35, 36	bitr4i 280	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑦}))
38	37	anbi2i 624	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ (𝐴 ∖ {𝑦})) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑦})))
39	33, 38	bitri 277	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) ↔ (𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑦})))
40		pm3.35 801	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 → 𝑥 ∈ {𝑦})) → 𝑥 ∈ {𝑦})
41	39, 40	sylbi 219	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴 ∖ (𝐴 ∖ {𝑦})) → 𝑥 ∈ {𝑦})
42	41	ssriv 3970	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝐴 ∖ {𝑦})) ⊆ {𝑦}
43		ssfi 8732	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ Fin ∧ (𝐴 ∖ (𝐴 ∖ {𝑦})) ⊆ {𝑦}) → (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin)
44	32, 42, 43	mp2an 690	. . . . . . . . . . 11 ⊢ (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin
45	44	a1i 11	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ (𝐴 ∖ {𝑦})) ∈ Fin)
46	26, 31, 45	elrabd 3681	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (𝐴 ∖ {𝑦}) ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin})
47	24, 46	nsyl3 140	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ¬ 𝑦 ∈ ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin})
48	47	eq0rdv 4356	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} = ∅)
49	48	sseq2d 3998	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (∩ 𝑓 ⊆ ∩ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ↔ ∩ 𝑓 ⊆ ∅))
50	21, 49	syl5ib 246	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → ∩ 𝑓 ⊆ ∅))
51		ss0 4351	. . . . 5 ⊢ (∩ 𝑓 ⊆ ∅ → ∩ 𝑓 = ∅)
52	50, 51	syl6 35	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → ∩ 𝑓 = ∅))
53	20, 52	jcad 515	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ({𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → (𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅)))
54	53	reximdv 3273	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → (∃𝑓 ∈ (UFil‘𝑋){𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∖ 𝑥) ∈ Fin} ⊆ 𝑓 → ∃𝑓 ∈ (UFil‘𝑋)(𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅)))
55	8, 54	mpd 15	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  {crab 3142   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  {csn 4560  ∩ cint 4868   class class class wbr 5058  ‘cfv 6349  ωcom 7574   ≼ cdom 8501  Fincfn 8503  Filcfil 22447  UFilcufil 22501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-ac2 9879

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rpss 7443  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-dju 9324  df-card 9362  df-ac 9536  df-fbas 20536  df-fg 20537  df-fil 22448  df-ufil 22503

This theorem is referenced by: (None)