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Mirrors > Home > MPE Home > Th. List > fin23lem19 | Structured version Visualization version GIF version |
Description: Lemma for fin23 10332. 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.) |
Ref | Expression |
---|---|
fin23lem.a | ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡) |
Ref | Expression |
---|---|
fin23lem19 | ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (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 | fin23lem12 10274 | . . . 4 ⊢ (𝐴 ∈ ω → (𝑈‘suc 𝐴) = if(((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅, (𝑈‘𝐴), ((𝑡‘𝐴) ∩ (𝑈‘𝐴)))) |
3 | eqif 4532 | . . . 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 4166 | . . . . 5 ⊢ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) | |
6 | ineq2 4171 | . . . . . . 7 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) | |
7 | 6 | eqeq1d 2739 | . . . . . 6 ⊢ ((𝑈‘suc 𝐴) = (𝑈‘𝐴) → (((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅ ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅)) |
8 | 7 | biimparc 481 | . . . . 5 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑡‘𝐴) ∩ (𝑈‘suc 𝐴)) = ∅) |
9 | 5, 8 | eqtrid 2789 | . . . 4 ⊢ ((((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅) |
10 | inss1 4193 | . . . . . 6 ⊢ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴) | |
11 | sseq1 3974 | . . . . . 6 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ↔ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) ⊆ (𝑡‘𝐴))) | |
12 | 10, 11 | mpbiri 258 | . . . . 5 ⊢ ((𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴)) |
13 | 12 | adantl 483 | . . . 4 ⊢ ((¬ ((𝑡‘𝐴) ∩ (𝑈‘𝐴)) = ∅ ∧ (𝑈‘suc 𝐴) = ((𝑡‘𝐴) ∩ (𝑈‘𝐴))) → (𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴)) |
14 | 9, 13 | orim12i 908 | . . 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 870 | 1 ⊢ (𝐴 ∈ ω → ((𝑈‘suc 𝐴) ⊆ (𝑡‘𝐴) ∨ ((𝑈‘suc 𝐴) ∩ (𝑡‘𝐴)) = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∩ cin 3914 ⊆ wss 3915 ∅c0 4287 ifcif 4491 ∪ cuni 4870 ran crn 5639 suc csuc 6324 ‘cfv 6501 ∈ cmpo 7364 ωcom 7807 seqωcseqom 8398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-seqom 8399 |
This theorem is referenced by: fin23lem20 10280 |
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