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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 6977 | . . . 4 ⊢ ∅ ∈ ω | |
3 | difsnpss 4279 | . . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω) | |
4 | 2, 3 | mpbi 219 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
5 | limom 6972 | . . . . 5 ⊢ Lim ω | |
6 | 5 | limenpsi 8020 | . . . 4 ⊢ (ω ∈ FinIV → ω ≈ (ω ∖ {∅})) |
7 | 6 | ensymd 7893 | . . 3 ⊢ (ω ∈ FinIV → (ω ∖ {∅}) ≈ ω) |
8 | fin4i 9003 | . . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ FinIV) | |
9 | 4, 7, 8 | sylancr 694 | . 2 ⊢ (ω ∈ FinIV → ¬ ω ∈ FinIV) |
10 | 1, 9 | pm2.65i 184 | 1 ⊢ ¬ ω ∈ FinIV |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 1977 ∖ cdif 3537 ⊊ wpss 3541 ∅c0 3874 {csn 4125 class class class wbr 4583 ωcom 6957 ≈ cen 7838 FinIVcfin4 8985 |
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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-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-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-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-om 6958 df-er 7629 df-en 7842 df-dom 7843 df-fin4 8992 |
This theorem is referenced by: infpssALT 9018 |
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