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Theorem fin23lem19 9809
 Description: Lemma for fin23 9862. 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 9804 . . . 4 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
3 eqif 4464 . . . 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 221 . . 3 (𝐴 ∈ ω → ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))))
5 incom 4108 . . . . 5 ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ((𝑡𝐴) ∩ (𝑈‘suc 𝐴))
6 ineq2 4113 . . . . . . 7 ((𝑈‘suc 𝐴) = (𝑈𝐴) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡𝐴) ∩ (𝑈𝐴)))
76eqeq1d 2760 . . . . . 6 ((𝑈‘suc 𝐴) = (𝑈𝐴) → (((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
87biimparc 483 . . . . 5 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
95, 8syl5eq 2805 . . . 4 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅)
10 inss1 4135 . . . . . 6 ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)
11 sseq1 3919 . . . . . 6 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)))
1210, 11mpbiri 261 . . . . 5 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
1312adantl 485 . . . 4 ((¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
149, 13orim12i 906 . . 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 868 1 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ifcif 4423  ∪ cuni 4801  ran crn 5529  suc csuc 6176  ‘cfv 6340   ∈ cmpo 7158  ωcom 7585  seqωcseqom 8099 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-seqom 8100 This theorem is referenced by:  fin23lem20  9810
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