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Theorem ominf4 10225

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 7829	. . . 4 ⊢ ∅ ∈ ω
3		difsnpss 4740	. . . 4 ⊢ (∅ ∈ ω ↔ (ω ∖ {∅}) ⊊ ω)
4	2, 3	mpbi 231	. . 3 ⊢ (ω ∖ {∅}) ⊊ ω
5		limom 7822	. . . . 5 ⊢ Lim ω
6	5	limenpsi 9080	. . . 4 ⊢ (ω ∈ Fin^IV → ω ≈ (ω ∖ {∅}))
7	6	ensymd 8942	. . 3 ⊢ (ω ∈ Fin^IV → (ω ∖ {∅}) ≈ ω)
8		fin4i 10211	. . 3 ⊢ (((ω ∖ {∅}) ⊊ ω ∧ (ω ∖ {∅}) ≈ ω) → ¬ ω ∈ Fin^IV)
9	4, 7, 8	sylancr 593	. 2 ⊢ (ω ∈ Fin^IV → ¬ ω ∈ Fin^IV)
10	1, 9	pm2.65i 195	1 ⊢ ¬ ω ∈ Fin^IV

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∈ wcel 2119 ∖ cdif 3880 ⊊ wpss 3884 ∅c0 4261 {csn 4555 class class class wbr 5072 ωcom 7806 ≈ cen 8880 Fin^IVcfin4 10193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-om 7807 df-er 8633 df-en 8884 df-dom 8885 df-fin4 10200

This theorem is referenced by: infpssALT 10226