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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 10305) 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 9095 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω) | |
2 | isfinite 9643 | . . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
3 | domtriom.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | eqid 2733 | . . . 4 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} | |
5 | fveq2 6888 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛)) | |
6 | fveq2 6888 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘)) | |
7 | 6 | cbviunv 5042 | . . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) |
8 | iuneq1 5012 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) | |
9 | 7, 8 | eqtrid 2785 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) |
10 | 5, 9 | difeq12d 4122 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗)) = ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
11 | 10 | cbvmptv 5260 | . . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
12 | 3, 4, 11 | domtriomlem 10433 | . . 3 ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴) |
13 | 2, 12 | sylnbir 331 | . 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 397 ∈ wcel 2107 {cab 2710 Vcvv 3475 ∖ cdif 3944 ⊆ wss 3947 𝒫 cpw 4601 ∪ ciun 4996 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 ωcom 7850 ≈ cen 8932 ≼ cdom 8933 ≺ csdm 8934 Fincfn 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cc 10426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 |
This theorem is referenced by: fin41 10435 dominf 10436 |
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