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Mirrors > Home > MPE Home > Th. List > domtriom | Structured version Visualization version GIF version |
Description: Trichotomy of equinumerosity for ω, proven using countable choice. Equivalently, all Dedekind-finite sets (as in isfin4-2 10347) are finite in the usual sense and conversely. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Ref | Expression |
---|---|
domtriom.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
domtriom | ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 9132 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω) | |
2 | isfinite 9685 | . . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
3 | domtriom.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | eqid 2728 | . . . 4 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} | |
5 | fveq2 6902 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛)) | |
6 | fveq2 6902 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘)) | |
7 | 6 | cbviunv 5047 | . . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) |
8 | iuneq1 5016 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) | |
9 | 7, 8 | eqtrid 2780 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) |
10 | 5, 9 | difeq12d 4123 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗)) = ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
11 | 10 | cbvmptv 5265 | . . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
12 | 3, 4, 11 | domtriomlem 10475 | . . 3 ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴) |
13 | 2, 12 | sylnbir 330 | . 2 ⊢ (¬ 𝐴 ≺ ω → ω ≼ 𝐴) |
14 | 1, 13 | impbii 208 | 1 ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 {cab 2705 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 𝒫 cpw 4606 ∪ ciun 5000 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 ωcom 7878 ≈ cen 8969 ≼ cdom 8970 ≺ csdm 8971 Fincfn 8972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-inf2 9674 ax-cc 10468 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-oadd 8499 df-er 8733 df-map 8855 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-dju 9934 df-card 9972 |
This theorem is referenced by: fin41 10477 dominf 10478 |
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