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Theorem ominf4 10291

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 23	. 2 ⊢ (ω ∈ Fin^IV → ω ∈ Fin^IV)
2		peano1 7881	. . . 4 ⊢ ∅ ∈ ω
3		difsnpss 4775	. . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω)
4	2, 3	mpbi 233	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
5		limom 7874	. . . . 5 ⊢ Lim ω
6	5	limenpsi 9136	. . . 4 ⊢ (ω ∈ Fin^IV → ω ≈ (ω ∖ {∅}))
7	6	ensymd 8998	. . 3 ⊢ (ω ∈ Fin^IV → (ω ∖ {∅}) ≈ ω)
8		fin4i 10277	. . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ Fin^IV)
9	4, 7, 8	sylancr 598	. 2 ⊢ (ω ∈ Fin^IV → ¬ ω ∈ Fin^IV)
10	1, 9	pm2.65i 196	1 ⊢ ¬ ω ∈ Fin^IV

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2143 ∖ cdif 3902 ⊊ wpss 3906 ∅c0 4286 {csn 4589 class class class wbr 5109 ωcom 7858 ≈ cen 8936 Fin^IVcfin4 10259

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-om 7859 df-er 8690 df-en 8940 df-dom 8941 df-fin4 10266

This theorem is referenced by: infpssALT 10292