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Mirrors > Home > MPE Home > Th. List > ominf4 | Structured version Visualization version GIF version |
Description: ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.) |
Ref | Expression |
---|---|
ominf4 | ⊢ ¬ ω ∈ FinIV |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (ω ∈ FinIV → ω ∈ FinIV) | |
2 | peano1 7875 | . . . 4 ⊢ ∅ ∈ ω | |
3 | difsnpss 4809 | . . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω) | |
4 | 2, 3 | mpbi 229 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
5 | limom 7867 | . . . . 5 ⊢ Lim ω | |
6 | 5 | limenpsi 9148 | . . . 4 ⊢ (ω ∈ FinIV → ω ≈ (ω ∖ {∅})) |
7 | 6 | ensymd 8997 | . . 3 ⊢ (ω ∈ FinIV → (ω ∖ {∅}) ≈ ω) |
8 | fin4i 10289 | . . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ FinIV) | |
9 | 4, 7, 8 | sylancr 587 | . 2 ⊢ (ω ∈ FinIV → ¬ ω ∈ FinIV) |
10 | 1, 9 | pm2.65i 193 | 1 ⊢ ¬ ω ∈ FinIV |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 ∖ cdif 3944 ⊊ wpss 3948 ∅c0 4321 {csn 4627 class class class wbr 5147 ωcom 7851 ≈ cen 8932 FinIVcfin4 10271 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-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-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-om 7852 df-er 8699 df-en 8936 df-dom 8937 df-fin4 10278 |
This theorem is referenced by: infpssALT 10304 |
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