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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 7111 | . 2 ⊢ ω ⊆ On | |
2 | 1 | sseli 3632 | 1 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2030 Oncon0 5761 ωcom 7107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-tr 4786 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-om 7108 |
This theorem is referenced by: nnoni 7114 nnord 7115 peano4 7130 findsg 7135 onasuc 7653 onmsuc 7654 nna0 7729 nnm0 7730 nnasuc 7731 nnmsuc 7732 nnesuc 7733 nnecl 7738 nnawordi 7746 nnmword 7758 nnawordex 7762 nnaordex 7763 oaabslem 7768 oaabs 7769 oaabs2 7770 omabslem 7771 omabs 7772 nnneo 7776 nneob 7777 onfin2 8193 findcard3 8244 dffi3 8378 card2inf 8501 elom3 8583 cantnfp1lem3 8615 cnfcomlem 8634 cnfcom 8635 cnfcom3 8639 finnum 8812 cardnn 8827 nnsdomel 8854 nnacda 9061 ficardun2 9063 ackbij1lem15 9094 ackbij2lem2 9100 ackbij2lem3 9101 ackbij2 9103 fin23lem22 9187 isf32lem5 9217 fin1a2lem4 9263 fin1a2lem9 9268 pwfseqlem3 9520 winainflem 9553 wunr1om 9579 tskr1om 9627 grothomex 9689 pion 9739 om2uzlt2i 12790 bnj168 30924 elhf2 32407 findreccl 32577 rdgeqoa 33348 finxpreclem4 33361 finxpreclem6 33363 harinf 37918 |
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