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Theorem fin23lem19 10376
Description: Lemma for fin23 10429. The first set in 𝑈 to see an input set is either contained in it or disjoint from it. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
Assertion
Ref Expression
fin23lem19 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem19
StepHypRef Expression
1 fin23lem.a . . . . 5 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
21fin23lem12 10371 . . . 4 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
3 eqif 4567 . . . 4 ((𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))) ↔ ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))))
42, 3sylib 218 . . 3 (𝐴 ∈ ω → ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))))
5 incom 4209 . . . . 5 ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ((𝑡𝐴) ∩ (𝑈‘suc 𝐴))
6 ineq2 4214 . . . . . . 7 ((𝑈‘suc 𝐴) = (𝑈𝐴) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡𝐴) ∩ (𝑈𝐴)))
76eqeq1d 2739 . . . . . 6 ((𝑈‘suc 𝐴) = (𝑈𝐴) → (((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
87biimparc 479 . . . . 5 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
95, 8eqtrid 2789 . . . 4 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅)
10 inss1 4237 . . . . . 6 ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)
11 sseq1 4009 . . . . . 6 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)))
1210, 11mpbiri 258 . . . . 5 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
1312adantl 481 . . . 4 ((¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
149, 13orim12i 909 . . 3 (((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))) → (((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅ ∨ (𝑈‘suc 𝐴) ⊆ (𝑡𝐴)))
154, 14syl 17 . 2 (𝐴 ∈ ω → (((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅ ∨ (𝑈‘suc 𝐴) ⊆ (𝑡𝐴)))
1615orcomd 872 1 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  c0 4333  ifcif 4525   cuni 4907  ran crn 5686  suc csuc 6386  cfv 6561  cmpo 7433  ωcom 7887  seqωcseqom 8487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488
This theorem is referenced by:  fin23lem20  10377
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