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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 400 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)) | |
2 | sdom2en01 9124 | . . . . 5 ⊢ (𝐴 ≺ 2𝑜 ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1𝑜)) | |
3 | 1, 2 | sylibr 224 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ≺ 2𝑜) |
4 | 3 | orcd 407 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
5 | onfin2 8152 | . . . . . . . 8 ⊢ ω = (On ∩ Fin) | |
6 | inss2 3834 | . . . . . . . 8 ⊢ (On ∩ Fin) ⊆ Fin | |
7 | 5, 6 | eqsstri 3635 | . . . . . . 7 ⊢ ω ⊆ Fin |
8 | 2onn 7720 | . . . . . . 7 ⊢ 2𝑜 ∈ ω | |
9 | 7, 8 | sselii 3600 | . . . . . 6 ⊢ 2𝑜 ∈ Fin |
10 | relsdom 7962 | . . . . . . 7 ⊢ Rel ≺ | |
11 | 10 | brrelexi 5158 | . . . . . 6 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → 𝐴 ∈ V) |
12 | fidomtri 8819 | . . . . . 6 ⊢ ((2𝑜 ∈ Fin ∧ 𝐴 ∈ V) → (2𝑜 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2𝑜)) | |
13 | 9, 11, 12 | sylancr 695 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → (2𝑜 ≼ 𝐴 ↔ ¬ 𝐴 ≺ 2𝑜)) |
14 | xp2cda 9002 | . . . . . . . . . 10 ⊢ (𝐴 ∈ V → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴)) | |
15 | 11, 14 | syl 17 | . . . . . . . . 9 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴)) |
16 | 15 | adantr 481 | . . . . . . . 8 ⊢ ((𝐴 ≺ (𝐴 +𝑐 𝐴) ∧ 2𝑜 ≼ 𝐴) → (𝐴 × 2𝑜) = (𝐴 +𝑐 𝐴)) |
17 | xpdom2g 8056 | . . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 2𝑜 ≼ 𝐴) → (𝐴 × 2𝑜) ≼ (𝐴 × 𝐴)) | |
18 | 11, 17 | sylan 488 | . . . . . . . 8 ⊢ ((𝐴 ≺ (𝐴 +𝑐 𝐴) ∧ 2𝑜 ≼ 𝐴) → (𝐴 × 2𝑜) ≼ (𝐴 × 𝐴)) |
19 | 16, 18 | eqbrtrrd 4677 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 +𝑐 𝐴) ∧ 2𝑜 ≼ 𝐴) → (𝐴 +𝑐 𝐴) ≼ (𝐴 × 𝐴)) |
20 | sdomdomtr 8093 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 +𝑐 𝐴) ∧ (𝐴 +𝑐 𝐴) ≼ (𝐴 × 𝐴)) → 𝐴 ≺ (𝐴 × 𝐴)) | |
21 | 19, 20 | syldan 487 | . . . . . 6 ⊢ ((𝐴 ≺ (𝐴 +𝑐 𝐴) ∧ 2𝑜 ≼ 𝐴) → 𝐴 ≺ (𝐴 × 𝐴)) |
22 | 21 | ex 450 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → (2𝑜 ≼ 𝐴 → 𝐴 ≺ (𝐴 × 𝐴))) |
23 | 13, 22 | sylbird 250 | . . . 4 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → (¬ 𝐴 ≺ 2𝑜 → 𝐴 ≺ (𝐴 × 𝐴))) |
24 | 23 | orrd 393 | . . 3 ⊢ (𝐴 ≺ (𝐴 +𝑐 𝐴) → (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
25 | 4, 24 | jaoi 394 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴)) → (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
26 | isfin5 9121 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 +𝑐 𝐴))) | |
27 | isfin6 9122 | . 2 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2𝑜 ∨ 𝐴 ≺ (𝐴 × 𝐴))) | |
28 | 25, 26, 27 | 3imtr4i 281 | 1 ⊢ (𝐴 ∈ FinV → 𝐴 ∈ FinVI) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ∅c0 3915 class class class wbr 4653 × cxp 5112 Oncon0 5723 (class class class)co 6650 ωcom 7065 1𝑜c1o 7553 2𝑜c2o 7554 ≈ cen 7952 ≼ cdom 7953 ≺ csdm 7954 Fincfn 7955 +𝑐 ccda 8989 FinVcfin5 9104 FinVIcfin6 9105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1o 7560 df-2o 7561 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-fin5 9111 df-fin6 9112 |
This theorem is referenced by: fin2so 33396 |
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