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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 CC. Equivalently, all Dedekind-finite sets (as in isfin4-2 9019) 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 7971 | . 2 ⊢ (ω ≼ 𝐴 → ¬ 𝐴 ≺ ω) | |
2 | isfinite 8432 | . . 3 ⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω) | |
3 | domtriom.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | eqid 2610 | . . . 4 ⊢ {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} = {𝑦 ∣ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ 𝒫 𝑛)} | |
5 | fveq2 6103 | . . . . . 6 ⊢ (𝑚 = 𝑛 → (𝑏‘𝑚) = (𝑏‘𝑛)) | |
6 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑏‘𝑗) = (𝑏‘𝑘)) | |
7 | 6 | cbviunv 4495 | . . . . . . 7 ⊢ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) |
8 | iuneq1 4470 | . . . . . . 7 ⊢ (𝑚 = 𝑛 → ∪ 𝑘 ∈ 𝑚 (𝑏‘𝑘) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) | |
9 | 7, 8 | syl5eq 2656 | . . . . . 6 ⊢ (𝑚 = 𝑛 → ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗) = ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘)) |
10 | 5, 9 | difeq12d 3691 | . . . . 5 ⊢ (𝑚 = 𝑛 → ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗)) = ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
11 | 10 | cbvmptv 4678 | . . . 4 ⊢ (𝑚 ∈ ω ↦ ((𝑏‘𝑚) ∖ ∪ 𝑗 ∈ 𝑚 (𝑏‘𝑗))) = (𝑛 ∈ ω ↦ ((𝑏‘𝑛) ∖ ∪ 𝑘 ∈ 𝑛 (𝑏‘𝑘))) |
12 | 3, 4, 11 | domtriomlem 9147 | . . 3 ⊢ (¬ 𝐴 ∈ Fin → ω ≼ 𝐴) |
13 | 2, 12 | sylnbir 320 | . 2 ⊢ (¬ 𝐴 ≺ ω → ω ≼ 𝐴) |
14 | 1, 13 | impbii 198 | 1 ⊢ (ω ≼ 𝐴 ↔ ¬ 𝐴 ≺ ω) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 𝒫 cpw 4108 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 ωcom 6957 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 Fincfn 7841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-cda 8873 |
This theorem is referenced by: fin41 9149 dominf 9150 |
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