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Mirrors > Home > ILE Home > Th. List > nnord | 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 4627 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | eloni 4393 | . 2 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝐴 ∈ ω → Ord 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 Ord word 4380 Oncon0 4381 ωcom 4607 |
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 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-uni 3825 df-int 3860 df-tr 4117 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 |
This theorem is referenced by: nnsucsssuc 6517 nnsucuniel 6520 nntri1 6521 nnsseleq 6526 nntr2 6528 phplem1 6880 phplem2 6881 phplem3 6882 phplem4 6883 phplem4dom 6890 nndomo 6892 dif1en 6907 nnwetri 6944 unsnfi 6947 ctmlemr 7137 nnnninf 7154 nnnninfeq 7156 nnnninfeq2 7157 nninfisol 7161 piord 7340 addnidpig 7365 archnqq 7446 frecfzennn 10457 hashp1i 10822 ennnfonelemk 12451 ennnfonelemg 12454 ennnfonelemhf1o 12464 ennnfonelemhom 12466 ctinfom 12479 nnsf 15216 peano4nninf 15217 nninfsellemeq 15225 |
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