Theorem fin23lem19 10333

Description: Lemma for fin23 10386. 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 10328	. . . 4 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
3		eqif 4568	. . . 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 217	. . 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 4200	. . . . 5 ⊢ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴))
6		ineq2 4205	. . . . . . 7 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
7	6	eqeq1d 2732	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅))
8	7	biimparc 478	. . . . 5 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
9	5, 8	eqtrid 2782	. . . 4 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)
10		inss1 4227	. . . . . 6 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)
11		sseq1 4006	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)))
12	10, 11	mpbiri 257	. . . . 5 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
13	12	adantl 480	. . . 4 ⊢ ((¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
14	9, 13	orim12i 905	. . 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 17	. 2 ⊢ (𝐴 ∈ ω → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ ∨ (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴)))
16	15	orcomd 867	1 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  ifcif 4527  ∪ cuni 4907  ran crn 5676  suc csuc 6365  ‘cfv 6542   ∈ cmpo 7413  ωcom 7857  seq_ωcseqom 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seqom 8450

This theorem is referenced by:  fin23lem20  10334