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Mirrors > Home > ILE Home > Th. List > ominf | GIF version |
Description: The set of natural numbers is not finite. Although we supply this theorem because we can, the more natural way to express "ω is infinite" is ω ≼ ω which is an instance of domrefg 6812. (Contributed by NM, 2-Jun-1998.) |
Ref | Expression |
---|---|
ominf | ⊢ ¬ ω ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4621 | . 2 ⊢ ω ∈ V | |
2 | domrefg 6812 | . 2 ⊢ (ω ∈ V → ω ≼ ω) | |
3 | infnfi 6942 | . 2 ⊢ (ω ≼ ω → ¬ ω ∈ Fin) | |
4 | 1, 2, 3 | mp2b 8 | 1 ⊢ ¬ ω ∈ Fin |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2164 Vcvv 2760 class class class wbr 4029 ωcom 4618 ≼ cdom 6784 Fincfn 6785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-tr 4128 df-id 4322 df-iord 4395 df-on 4397 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-er 6578 df-en 6786 df-dom 6787 df-fin 6788 |
This theorem is referenced by: inffiexmid 6953 |
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