Theorem fin23lem19 10363

Description: Lemma for fin23 10416. 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(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)

Assertion

Ref	Expression
fin23lem19	⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))

Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢

Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem19

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	fin23lem12 10358	. . . 4 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
3		eqif 4524	. . . 4 ⊢ ((𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) ↔ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) ∨ (¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))))
4	2, 3	sylib 221	. . 3 ⊢ (𝐴 ∈ ω → ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) ∨ (¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))))
5		incom 4155	. . . . 5 ⊢ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴))
6		ineq2 4160	. . . . . . 7 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
7	6	eqeq1d 2762	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅))
8	7	biimparc 485	. . . . 5 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
9	5, 8	eqtrid 2807	. . . 4 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)
10		inss1 4182	. . . . . 6 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)
11		sseq1 3956	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)))
12	10, 11	mpbiri 261	. . . . 5 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
13	12	adantl 487	. . . 4 ⊢ ((¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
14	9, 13	orim12i 922	. . 3 ⊢ (((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) ∨ (¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ ∨ (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴)))
15	4, 14	syl 18	. 2 ⊢ (𝐴 ∈ ω → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ ∨ (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴)))
16	15	orcomd 885	1 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3450   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  ifcif 4482  ∪ cuni 4867  ran crn 5656  suc csuc 6361  ‘cfv 6535   ∈ cmpo 7418  ωcom 7868  seq_ωcseqom 8443

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-seqom 8444

This theorem is used by:  fin23lem20  10364