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Theorem fin23lem19 9041
 Description: Lemma for fin23 9094. 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 9036 . . . 4 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
3 eqif 4076 . . . 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 207 . . 3 (𝐴 ∈ ω → ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))))
5 incom 3767 . . . . 5 ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ((𝑡𝐴) ∩ (𝑈‘suc 𝐴))
6 ineq2 3770 . . . . . . 7 ((𝑈‘suc 𝐴) = (𝑈𝐴) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡𝐴) ∩ (𝑈𝐴)))
76eqeq1d 2612 . . . . . 6 ((𝑈‘suc 𝐴) = (𝑈𝐴) → (((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
87biimparc 503 . . . . 5 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
95, 8syl5eq 2656 . . . 4 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅)
10 inss1 3795 . . . . . 6 ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)
11 sseq1 3589 . . . . . 6 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)))
1210, 11mpbiri 247 . . . . 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 537 . . 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 402 1 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ∪ cuni 4372  ran crn 5039  suc csuc 5642  ‘cfv 5804   ↦ cmpt2 6551  ωcom 6957  seq𝜔cseqom 7429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430 This theorem is referenced by:  fin23lem20  9042
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