![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 867 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
2 | sdom2en01 10339 | . . . . 5 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
3 | 1, 2 | sylibr 234 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ≺ 2o) |
4 | 3 | orcd 873 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
5 | onfin2 9265 | . . . . . . . 8 ⊢ ω = (On ∩ Fin) | |
6 | inss2 4245 | . . . . . . . 8 ⊢ (On ∩ Fin) ⊆ Fin | |
7 | 5, 6 | eqsstri 4029 | . . . . . . 7 ⊢ ω ⊆ Fin |
8 | 2onn 8678 | . . . . . . 7 ⊢ 2o ∈ ω | |
9 | 7, 8 | sselii 3991 | . . . . . 6 ⊢ 2o ∈ Fin |
10 | relsdom 8990 | . . . . . . 7 ⊢ Rel ≺ | |
11 | 10 | brrelex1i 5744 | . . . . . 6 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
12 | fidomtri 10030 | . . . . . 6 ⊢ ((2o ∈ Fin ∧ 𝐴 ∈ V) → (2o ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2o)) | |
13 | 9, 11, 12 | sylancr 587 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (2o ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2o)) |
14 | xp2dju 10214 | . . . . . . . 8 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | |
15 | xpdom1g 9107 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 2o ≼ 𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴)) | |
16 | 11, 15 | sylan 580 | . . . . . . . 8 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (2o × 𝐴) ≼ (𝐴 × 𝐴)) |
17 | 14, 16 | eqbrtrrid 5183 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) |
18 | sdomdomtr 9148 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≺ (𝐴 × 𝐴)) | |
19 | 17, 18 | syldan 591 | . . . . . 6 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → 𝐴 ≺ (𝐴 × 𝐴)) |
20 | 19 | ex 412 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (2o ≼ 𝐴 → 𝐴 ≺ (𝐴 × 𝐴))) |
21 | 13, 20 | sylbird 260 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (¬ 𝐴 ≺ 2o → 𝐴 ≺ (𝐴 × 𝐴))) |
22 | 21 | orrd 863 | . . 3 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
23 | 4, 22 | jaoi 857 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
24 | isfin5 10336 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
25 | isfin6 10337 | . 2 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | |
26 | 23, 24, 25 | 3imtr4i 292 | 1 ⊢ (𝐴 ∈ FinV → 𝐴 ∈ FinVI) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ∅c0 4338 class class class wbr 5147 × cxp 5686 Oncon0 6385 ωcom 7886 1oc1o 8497 2oc2o 8498 ≈ cen 8980 ≼ cdom 8981 ≺ csdm 8982 Fincfn 8983 ⊔ cdju 9935 FinVcfin5 10319 FinVIcfin6 10320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-om 7887 df-1st 8012 df-2nd 8013 df-1o 8504 df-2o 8505 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-fin5 10326 df-fin6 10327 |
This theorem is referenced by: fin2so 37593 |
Copyright terms: Public domain | W3C validator |