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Mirrors > Home > MPE Home > Th. List > nnord | Structured version Visualization version GIF version |
Description: A natural number is ordinal. (Contributed by NM, 17-Oct-1995.) |
Ref | Expression |
---|---|
nnord | ⊢ (𝐴 ∈ ω → Ord 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7575 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | eloni 6194 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐴 ∈ ω → Ord 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Ord word 6183 Oncon0 6184 ωcom 7569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 |
This theorem is referenced by: nnlim 7582 nnsuc 7586 omsucne 7587 nnaordi 8233 nnaord 8234 nnaword 8242 nnmord 8247 nnmwordi 8250 nnawordex 8252 omsmo 8270 phplem1 8684 phplem2 8685 phplem3 8686 phplem4 8687 php 8689 php4 8692 nndomo 8700 ominf 8718 isinf 8719 pssnn 8724 dif1en 8739 unblem1 8758 isfinite2 8764 unfilem1 8770 inf3lem5 9083 inf3lem6 9084 cantnfp1lem2 9130 cantnfp1lem3 9131 dif1card 9424 pwsdompw 9614 ackbij1lem5 9634 ackbij1lem14 9643 ackbij1lem16 9645 ackbij1b 9649 ackbij2 9653 sornom 9687 infpssrlem4 9716 infpssrlem5 9717 fin23lem26 9735 fin23lem23 9736 isf32lem2 9764 isf32lem3 9765 isf32lem4 9766 domtriomlem 9852 axdc3lem2 9861 axdc3lem4 9863 canthp1lem2 10063 elni2 10287 piord 10290 addnidpi 10311 indpi 10317 om2uzf1oi 13309 fzennn 13324 hashp1i 13752 bnj529 31911 bnj1098 31954 bnj570 32076 bnj594 32083 bnj580 32084 bnj967 32116 bnj1001 32129 bnj1053 32145 bnj1071 32146 hfun 33536 finminlem 33563 finxpsuclem 34560 finxpsuc 34561 wepwso 39521 nndomog 39775 |
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