Theorem fin23lem19 9196

Description: Lemma for fin23 9249. 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 9191	. . . 4 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴))))
3		eqif 4159	. . . 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 208	. . 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 3838	. . . . 5 ⊢ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴))
6		ineq2 3841	. . . . . . 7 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))
7	6	eqeq1d 2653	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅))
8	7	biimparc 503	. . . . 5 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅)
9	5, 8	syl5eq 2697	. . . 4 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)
10		inss1 3866	. . . . . 6 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)
11		sseq1 3659	. . . . . 6 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴)))
12	10, 11	mpbiri 248	. . . . 5 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
13	12	adantl 481	. . . 4 ⊢ ((¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴))
14	9, 13	orim12i 537	. . 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 402	1 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∩ cin 3606   ⊆ wss 3607  ∅c0 3948  ifcif 4119  ∪ cuni 4468  ran crn 5144  suc csuc 5763  ‘cfv 5926   ↦ cmpt2 6692  ωcom 7107  seq_𝜔cseqom 7587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seqom 7588

This theorem is referenced by:  fin23lem20  9197