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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 865 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
2 | sdom2en01 10279 | . . . . 5 ⊢ (𝐴 ≺ 2o ↔ (𝐴 = ∅ ∨ 𝐴 ≈ 1o)) | |
3 | 1, 2 | sylibr 233 | . . . 4 ⊢ (𝐴 = ∅ → 𝐴 ≺ 2o) |
4 | 3 | orcd 871 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
5 | onfin2 9214 | . . . . . . . 8 ⊢ ω = (On ∩ Fin) | |
6 | inss2 4225 | . . . . . . . 8 ⊢ (On ∩ Fin) ⊆ Fin | |
7 | 5, 6 | eqsstri 4012 | . . . . . . 7 ⊢ ω ⊆ Fin |
8 | 2onn 8624 | . . . . . . 7 ⊢ 2o ∈ ω | |
9 | 7, 8 | sselii 3975 | . . . . . 6 ⊢ 2o ∈ Fin |
10 | relsdom 8929 | . . . . . . 7 ⊢ Rel ≺ | |
11 | 10 | brrelex1i 5724 | . . . . . 6 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → 𝐴 ∈ V) |
12 | fidomtri 9970 | . . . . . 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 10153 | . . . . . . . 8 ⊢ (2o × 𝐴) = (𝐴 ⊔ 𝐴) | |
15 | xpdom1g 9052 | . . . . . . . . 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 5177 | . . . . . . 7 ⊢ ((𝐴 ≺ (𝐴 ⊔ 𝐴) ∧ 2o ≼ 𝐴) → (𝐴 ⊔ 𝐴) ≼ (𝐴 × 𝐴)) |
18 | sdomdomtr 9093 | . . . . . . 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 413 | . . . . 5 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (2o ≼ 𝐴 → 𝐴 ≺ (𝐴 × 𝐴))) |
21 | 13, 20 | sylbird 259 | . . . 4 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (¬ 𝐴 ≺ 2o → 𝐴 ≺ (𝐴 × 𝐴))) |
22 | 21 | orrd 861 | . . 3 ⊢ (𝐴 ≺ (𝐴 ⊔ 𝐴) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
23 | 4, 22 | jaoi 855 | . 2 ⊢ ((𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴)) → (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) |
24 | isfin5 10276 | . 2 ⊢ (𝐴 ∈ FinV ↔ (𝐴 = ∅ ∨ 𝐴 ≺ (𝐴 ⊔ 𝐴))) | |
25 | isfin6 10277 | . 2 ⊢ (𝐴 ∈ FinVI ↔ (𝐴 ≺ 2o ∨ 𝐴 ≺ (𝐴 × 𝐴))) | |
26 | 23, 24, 25 | 3imtr4i 291 | 1 ⊢ (𝐴 ∈ FinV → 𝐴 ∈ FinVI) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 Vcvv 3473 ∩ cin 3943 ∅c0 4318 class class class wbr 5141 × cxp 5667 Oncon0 6353 ωcom 7838 1oc1o 8441 2oc2o 8442 ≈ cen 8919 ≼ cdom 8920 ≺ csdm 8921 Fincfn 8922 ⊔ cdju 9875 FinVcfin5 10259 FinVIcfin6 10260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-om 7839 df-1st 7957 df-2nd 7958 df-1o 8448 df-2o 8449 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-dju 9878 df-card 9916 df-fin5 10266 df-fin6 10267 |
This theorem is referenced by: fin2so 36277 |
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