Theorem fin23lem20 10024

Description: Lemma for fin23 10076. 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 8256	. . . 4 ⊢ 𝑈 Fn ω
3		peano2 7711	. . . 4 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
4		fnfvelrn 6940	. . . 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 4891	. . 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 10023	. 2 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))
9		sstr2 3924	. . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) → ∩ ran 𝑈 ⊆ (𝑡‘𝐴)))
10		ssdisj 4390	. . . 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 961	. 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 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  ∪ cuni 4836  ∩ cint 4876  ran crn 5581  suc csuc 6253   Fn wfn 6413  ‘cfv 6418   ∈ cmpo 7257  ωcom 7687  seq_ωcseqom 8248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seqom 8249

This theorem is referenced by:  fin23lem30  10029