Theorem fin23lem19 10335

Description: Lemma for fin23 10388. 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 10330	. . . 4 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
3		eqif 4531	. . . 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 4162	. . . . 5 ⊢ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴))
6		ineq2 4167	. . . . . . 7 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
7	6	eqeq1d 2767	. . . . . 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 2812	. . . 4 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)
10		inss1 4189	. . . . . 6 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)
11		sseq1 3963	. . . . . 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 2146  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286  ifcif 4489  ∪ cuni 4874  ran crn 5664  suc csuc 6366  ‘cfv 6540   ∈ cmpo 7421  ωcom 7868  seq_ωcseqom 8440

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seqom 8441

This theorem is used by:  fin23lem20  10336