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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 10354) 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 9139 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω) | |
| 2 | isfinite 9692 | . . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
| 3 | domtriom.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 4 | eqid 2737 | . . . 4 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} | |
| 5 | fveq2 6906 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛)) | |
| 6 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘)) | |
| 7 | 6 | cbviunv 5040 | . . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) |
| 8 | iuneq1 5008 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) | |
| 9 | 7, 8 | eqtrid 2789 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) |
| 10 | 5, 9 | difeq12d 4127 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗)) = ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
| 11 | 10 | cbvmptv 5255 | . . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
| 12 | 3, 4, 11 | domtriomlem 10482 | . . 3 ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴) |
| 13 | 2, 12 | sylnbir 331 | . 2 ⊢ (¬ 𝐴 ≺ ω → ω ≼ 𝐴) |
| 14 | 1, 13 | impbii 209 | 1 ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 {cab 2714 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 𝒫 cpw 4600 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 ωcom 7887 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 |
| 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-rmo 3380 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-iun 4993 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-pred 6321 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-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 |
| This theorem is referenced by: fin41 10484 dominf 10485 |
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