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| Mirrors > Home > MPE Home > Th. List > fin23lem20 | Structured version Visualization version GIF version | ||
| Description: Lemma for fin23 10429. 𝑋 is either contained in or disjoint from all input sets. (Contributed by Stefan O'Rear, 1-Nov-2014.) |
| Ref | Expression |
|---|---|
| fin23lem.a | ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) |
| Ref | Expression |
|---|---|
| fin23lem20 | ⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)) |
| 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 | fnseqom 8495 | . . . 4 ⊢ 𝑈 Fn ω |
| 3 | peano2 7912 | . . . 4 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω) | |
| 4 | fnfvelrn 7100 | . . . 4 ⊢ ((𝑈 Fn ω ∧ suc 𝐴 ∈ ω) → (𝑈‘suc 𝐴) ∈ ran 𝑈) | |
| 5 | 2, 3, 4 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) ∈ ran 𝑈) |
| 6 | intss1 4963 | . . 3 ⊢ ((𝑈‘suc 𝐴) ∈ ran 𝑈 → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴)) | |
| 7 | 5, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ω → ∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴)) |
| 8 | 1 | fin23lem19 10376 | . 2 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)) |
| 9 | sstr2 3990 | . . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) → ∩ ran 𝑈 ⊆ (𝑡‘𝐴))) | |
| 10 | ssdisj 4460 | . . . 4 ⊢ ((∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) ∧ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅) | |
| 11 | 10 | ex 412 | . . 3 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅ → (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)) |
| 12 | 9, 11 | orim12d 967 | . 2 ⊢ (∩ ran 𝑈 ⊆ (𝑈‘suc 𝐴) → (((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅))) |
| 13 | 7, 8, 12 | sylc 65 | 1 ⊢ (𝐴 ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘𝐴) ∨ (∩ ran 𝑈 ∩ (𝑡‘𝐴)) = ∅)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∨ wo 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 ∪ cuni 4907 ∩ cint 4946 ran crn 5686 suc csuc 6386 Fn wfn 6556 ‘cfv 6561 ∈ cmpo 7433 ωcom 7887 seqωcseqom 8487 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 |
| This theorem is referenced by: fin23lem30 10382 |
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