ominf4 - Metamath Proof Explorer

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Theorem ominf4 10206

Description: ω is Dedekind infinite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Proof shortened by Mario Carneiro, 16-May-2015.)

**Assertion**
Ref	Expression
ominf4	⊢ ¬ ω ∈ Fin^IV

**Proof of Theorem ominf4**
Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (ω ∈ Fin^IV → ω ∈ Fin^IV)
2		peano1 7822	. . . 4 ⊢ ∅ ∈ ω
3		difsnpss 4758	. . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω)
4	2, 3	mpbi 230	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
5		limom 7815	. . . . 5 ⊢ Lim ω
6	5	limenpsi 9069	. . . 4 ⊢ (ω ∈ Fin^IV → ω ≈ (ω ∖ {∅}))
7	6	ensymd 8930	. . 3 ⊢ (ω ∈ Fin^IV → (ω ∖ {∅}) ≈ ω)
8		fin4i 10192	. . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ Fin^IV)
9	4, 7, 8	sylancr 587	. 2 ⊢ (ω ∈ Fin^IV → ¬ ω ∈ Fin^IV)
10	1, 9	pm2.65i 194	1 ⊢ ¬ ω ∈ Fin^IV

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2109 ∖ cdif 3900 ⊊ wpss 3904 ∅c0 4284 {csn 4577 class class class wbr 5092 ωcom 7799 ≈ cen 8869 Fin^IVcfin4 10174

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-om 7800 df-er 8625 df-en 8873 df-dom 8874 df-fin4 10181

This theorem is referenced by: infpssALT 10207