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Theorem fin23lem19 9747
Description: Lemma for fin23 9800. 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 9742 . . . 4 (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡𝐴) ∩ (𝑈𝐴)) = ∅, (𝑈𝐴), ((𝑡𝐴) ∩ (𝑈𝐴))))
3 eqif 4510 . . . 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 219 . . 3 (𝐴 ∈ ω → ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) ∨ (¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)))))
5 incom 4182 . . . . 5 ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ((𝑡𝐴) ∩ (𝑈‘suc 𝐴))
6 ineq2 4187 . . . . . . 7 ((𝑈‘suc 𝐴) = (𝑈𝐴) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡𝐴) ∩ (𝑈𝐴)))
76eqeq1d 2828 . . . . . 6 ((𝑈‘suc 𝐴) = (𝑈𝐴) → (((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅))
87biimparc 480 . . . . 5 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑡𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
95, 8syl5eq 2873 . . . 4 ((((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅)
10 inss1 4209 . . . . . 6 ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)
11 sseq1 3996 . . . . . 6 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ↔ ((𝑡𝐴) ∩ (𝑈𝐴)) ⊆ (𝑡𝐴)))
1210, 11mpbiri 259 . . . . 5 ((𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
1312adantl 482 . . . 4 ((¬ ((𝑡𝐴) ∩ (𝑈𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡𝐴) ∩ (𝑈𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡𝐴))
149, 13orim12i 904 . . 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 867 1 (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡𝐴)) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843   = wceq 1530  wcel 2107  Vcvv 3500  cin 3939  wss 3940  c0 4295  ifcif 4470   cuni 4837  ran crn 5555  suc csuc 6191  cfv 6352  cmpo 7150  ωcom 7568  seqωcseqom 8074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-seqom 8075
This theorem is referenced by:  fin23lem20  9748
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