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Mirrors > Home > MPE Home > Th. List > fin56 | Structured version Visualization version GIF version |
Description: Every V-finite set is VI-finite because multiplication dominates addition for cardinals. (Contributed by Stefan O'Rear, 29-Oct-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
fin56 | ⊢ (𝐴 ∈ FinV → 𝐴 ∈ FinVI) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 863 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
2 | sdom2en01 9718 | . . . . 5 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
3 | 1, 2 | sylibr 236 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ≺ 2o) |
4 | 3 | orcd 869 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
5 | onfin2 8704 | . . . . . . . 8 ⊢ ω = (On ∩ Fin) | |
6 | inss2 4205 | . . . . . . . 8 ⊢ (On ∩ Fin) ⊆ Fin | |
7 | 5, 6 | eqsstri 4000 | . . . . . . 7 ⊢ ω ⊆ Fin |
8 | 2onn 8260 | . . . . . . 7 ⊢ 2o ∈ ω | |
9 | 7, 8 | sselii 3963 | . . . . . 6 ⊢ 2o ∈ Fin |
10 | relsdom 8510 | . . . . . . 7 ⊢ Rel ≺ | |
11 | 10 | brrelex1i 5602 | . . . . . 6 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
12 | fidomtri 9416 | . . . . . 6 ⊢ ((2o ∈ Fin ∧ 𝐴 ∈ V) → (2o ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2o)) | |
13 | 9, 11, 12 | sylancr 589 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (2o ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2o)) |
14 | xp2dju 9596 | . . . . . . . 8 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | |
15 | xpdom1g 8608 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 2o ≼ 𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴)) | |
16 | 11, 15 | sylan 582 | . . . . . . . 8 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴)) |
17 | 14, 16 | eqbrtrrid 5094 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) |
18 | sdomdomtr 8644 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≺ (𝐴 × 𝐴)) | |
19 | 17, 18 | syldan 593 | . . . . . 6 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → 𝐴 ≺ (𝐴 × 𝐴)) |
20 | 19 | ex 415 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (2o ≼ 𝐴 → 𝐴 ≺ (𝐴 × 𝐴))) |
21 | 13, 20 | sylbird 262 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (¬ 𝐴 ≺ 2o → 𝐴 ≺ (𝐴 × 𝐴))) |
22 | 21 | orrd 859 | . . 3 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
23 | 4, 22 | jaoi 853 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
24 | isfin5 9715 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
25 | isfin6 9716 | . 2 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | |
26 | 23, 24, 25 | 3imtr4i 294 | 1 ⊢ (𝐴 ∈ FinV → 𝐴 ∈ FinVI) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∩ cin 3934 ∅c0 4290 class class class wbr 5058 × cxp 5547 Oncon0 6185 ωcom 7574 1oc1o 8089 2oc2o 8090 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 Fincfn 8503 ⊔ cdju 9321 FinVcfin5 9698 FinVIcfin6 9699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1st 7683 df-2nd 7684 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-fin5 9705 df-fin6 9706 |
This theorem is referenced by: fin2so 34873 |
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