Theorem fin23lem20 10339

Description: Lemma for fin23 10391. 𝑋 is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.)

Hypothesis

Ref	Expression
fin23lem.a	⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)

Assertion

Ref	Expression
fin23lem20	⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))

Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢

Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem20

Step	Hyp	Ref	Expression
1		fin23lem.a	. . . . 5 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
2	1	fnseqom 8451	. . . 4 ⊢ 𝑈 Fn ω
3		peano2 7895	. . . 4 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
4		fnfvelrn 7082	. . . 4 ⊢ ((𝑈 Fn ω ∧ suc 𝐴 ∈ ω) → (𝑈‘suc 𝐴) ∈ ran 𝑈)
5	2, 3, 4	sylancr 599	. . 3 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) ∈ ran 𝑈)
6		intss1 4933	. . 3 ⊢ ((𝑈‘suc 𝐴) ∈ ran 𝑈 → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴))
7	5, 6	syl 18	. 2 ⊢ (𝐴 ∈ ω → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴))
8	1	fin23lem19 10338	. 2 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))
9		sstr2 3947	. . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) → ∩ ran 𝑈 ⊆ (𝑡‘𝐴)))
10		ssdisj 4423	. . . 4 ⊢ ((∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) ∧ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)
11	10	ex 418	. . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))
12	9, 11	orim12d 979	. 2 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)))
13	7, 8, 12	sylc 66	1 ⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2146  Vcvv 3458   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  ifcif 4492  ∪ cuni 4877  ∩ cint 4917  ran crn 5667  suc csuc 6369   Fn wfn 6538  ‘cfv 6543   ∈ cmpo 7425  ωcom 7871  seq_ωcseqom 8443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-seqom 8444

This theorem is used by:  fin23lem30  10344