Theorem fin23lem20 10406

Description: Lemma for fin23 10458. 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 8511	. . . 4 ⊢ 𝑈 Fn ω
3		peano2 7929	. . . 4 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
4		fnfvelrn 7114	. . . 4 ⊢ ((𝑈 Fn ω ∧ suc 𝐴 ∈ ω) → (𝑈‘suc 𝐴) ∈ ran 𝑈)
5	2, 3, 4	sylancr 586	. . 3 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) ∈ ran 𝑈)
6		intss1 4987	. . 3 ⊢ ((𝑈‘suc 𝐴) ∈ ran 𝑈 → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴))
7	5, 6	syl 17	. 2 ⊢ (𝐴 ∈ ω → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴))
8	1	fin23lem19 10405	. 2 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))
9		sstr2 4015	. . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) → ∩ ran 𝑈 ⊆ (𝑡‘𝐴)))
10		ssdisj 4483	. . . 4 ⊢ ((∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) ∧ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)
11	10	ex 412	. . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))
12	9, 11	orim12d 965	. 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 65	1 ⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ifcif 4548  ∪ cuni 4931  ∩ cint 4970  ran crn 5701  suc csuc 6397   Fn wfn 6568  ‘cfv 6573   ∈ cmpo 7450  ωcom 7903  seq_ωcseqom 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-seqom 8504

This theorem is referenced by:  fin23lem30  10411