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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 868 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
| 2 | sdom2en01 10342 | . . . . 5 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
| 3 | 1, 2 | sylibr 234 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ≺ 2o) |
| 4 | 3 | orcd 874 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
| 5 | onfin2 9268 | . . . . . . . 8 ⊢ ω = (On ∩ Fin) | |
| 6 | inss2 4238 | . . . . . . . 8 ⊢ (On ∩ Fin) ⊆ Fin | |
| 7 | 5, 6 | eqsstri 4030 | . . . . . . 7 ⊢ ω ⊆ Fin |
| 8 | 2onn 8680 | . . . . . . 7 ⊢ 2o ∈ ω | |
| 9 | 7, 8 | sselii 3980 | . . . . . 6 ⊢ 2o ∈ Fin |
| 10 | relsdom 8992 | . . . . . . 7 ⊢ Rel ≺ | |
| 11 | 10 | brrelex1i 5741 | . . . . . 6 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
| 12 | fidomtri 10033 | . . . . . 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 10217 | . . . . . . . 8 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | |
| 15 | xpdom1g 9109 | . . . . . . . . 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 5179 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) |
| 18 | sdomdomtr 9150 | . . . . . . 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 864 | . . 3 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
| 23 | 4, 22 | jaoi 858 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
| 24 | isfin5 10339 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
| 25 | isfin6 10340 | . 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 848 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 × cxp 5683 Oncon0 6384 ωcom 7887 1oc1o 8499 2oc2o 8500 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 ⊔ cdju 9938 FinVcfin5 10322 FinVIcfin6 10323 |
| 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-pow 5365 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-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-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-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-fin5 10329 df-fin6 10330 |
| This theorem is referenced by: fin2so 37614 |
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