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Mirrors > Home > MPE Home > Th. List > nnon | Structured version Visualization version GIF version |
Description: A natural number is an ordinal number. (Contributed by NM, 27-Jun-1994.) |
Ref | Expression |
---|---|
nnon | ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 7573 | . 2 ⊢ ω ⊆ On | |
2 | 1 | sseli 3960 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 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: nnoni 7576 nnord 7577 peano4 7593 findsg 7598 onasuc 8142 onmsuc 8143 nna0 8219 nnm0 8220 nnasuc 8221 nnmsuc 8222 nnesuc 8223 nnecl 8228 nnawordi 8236 nnmword 8248 nnawordex 8252 nnaordex 8253 oaabslem 8259 oaabs 8260 oaabs2 8261 omabslem 8262 omabs 8263 nnneo 8267 nneob 8268 onfin2 8698 findcard3 8749 dffi3 8883 card2inf 9007 elom3 9099 cantnfp1lem3 9131 cnfcomlem 9150 cnfcom 9151 cnfcom3 9155 finnum 9365 cardnn 9380 nnsdomel 9407 nnadju 9611 ficardun2 9613 ackbij1lem15 9644 ackbij2lem2 9650 ackbij2lem3 9651 ackbij2 9653 fin23lem22 9737 isf32lem5 9767 fin1a2lem4 9813 fin1a2lem9 9818 pwfseqlem3 10070 winainflem 10103 wunr1om 10129 tskr1om 10177 grothomex 10239 pion 10289 om2uzlt2i 13307 bnj168 31899 satfvsuc 32505 satf0suc 32520 sat1el2xp 32523 fmlasuc0 32528 elhf2 33533 findreccl 33698 rdgeqoa 34533 exrecfnlem 34542 finxpreclem4 34557 finxpreclem6 34559 harinf 39509 harsucnn 39781 |
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