ominf4 - Metamath Proof Explorer

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Theorem ominf4 10306

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 7875	. . . 4 ⊢ ∅ ∈ ω
3		difsnpss 4805	. . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω)
4	2, 3	mpbi 229	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
5		limom 7867	. . . . 5 ⊢ Lim ω
6	5	limenpsi 9151	. . . 4 ⊢ (ω ∈ Fin^IV → ω ≈ (ω ∖ {∅}))
7	6	ensymd 9000	. . 3 ⊢ (ω ∈ Fin^IV → (ω ∖ {∅}) ≈ ω)
8		fin4i 10292	. . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ Fin^IV)
9	4, 7, 8	sylancr 586	. 2 ⊢ (ω ∈ Fin^IV → ¬ ω ∈ Fin^IV)
10	1, 9	pm2.65i 193	1 ⊢ ¬ ω ∈ Fin^IV

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2098 ∖ cdif 3940 ⊊ wpss 3944 ∅c0 4317 {csn 4623 class class class wbr 5141 ωcom 7851 ≈ cen 8935 Fin^IVcfin4 10274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7852 df-er 8702 df-en 8939 df-dom 8940 df-fin4 10281

This theorem is referenced by: infpssALT 10307