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Theorem ominf4 10343

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 7900	. . . 4 ⊢ ∅ ∈ ω
3		difsnpss 4815	. . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω)
4	2, 3	mpbi 229	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
5		limom 7892	. . . . 5 ⊢ Lim ω
6	5	limenpsi 9183	. . . 4 ⊢ (ω ∈ Fin^IV → ω ≈ (ω ∖ {∅}))
7	6	ensymd 9032	. . 3 ⊢ (ω ∈ Fin^IV → (ω ∖ {∅}) ≈ ω)
8		fin4i 10329	. . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ Fin^IV)
9	4, 7, 8	sylancr 585	. 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 3946 ⊊ wpss 3950 ∅c0 4326 {csn 4632 class class class wbr 5152 ωcom 7876 ≈ cen 8967 Fin^IVcfin4 10311

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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-om 7877 df-er 8731 df-en 8971 df-dom 8972 df-fin4 10318

This theorem is referenced by: infpssALT 10344