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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 6784. (Contributed by NM, 2-Jun-1998.) |
Ref | Expression |
---|---|
ominf | ⊢ ¬ ω ∈ Fin |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4606 | . 2 ⊢ ω ∈ V | |
2 | domrefg 6784 | . 2 ⊢ (ω ∈ V → ω ≼ ω) | |
3 | infnfi 6912 | . 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 2159 Vcvv 2751 class class class wbr 4017 ωcom 4603 ≼ cdom 6756 Fincfn 6757 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 ax-iinf 4601 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-sbc 2977 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-br 4018 df-opab 4079 df-tr 4116 df-id 4307 df-iord 4380 df-on 4382 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-rn 4651 df-res 4652 df-ima 4653 df-iota 5192 df-fun 5232 df-fn 5233 df-f 5234 df-f1 5235 df-fo 5236 df-f1o 5237 df-fv 5238 df-er 6552 df-en 6758 df-dom 6759 df-fin 6760 |
This theorem is referenced by: inffiexmid 6923 |
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