![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4531 | . 2 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | eloni 4305 | . 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 1481 Ord word 4292 Oncon0 4293 ωcom 4512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-uni 3745 df-int 3780 df-tr 4035 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 |
This theorem is referenced by: nnsucsssuc 6396 nnsucuniel 6399 nntri1 6400 nnsseleq 6405 nntr2 6407 phplem1 6754 phplem2 6755 phplem3 6756 phplem4 6757 phplem4dom 6764 nndomo 6766 dif1en 6781 nnwetri 6812 unsnfi 6815 ctmlemr 7001 nnnninf 7031 piord 7143 addnidpig 7168 archnqq 7249 frecfzennn 10230 hashp1i 10588 ennnfonelemk 11949 ennnfonelemg 11952 ennnfonelemhf1o 11962 ennnfonelemhom 11964 ctinfom 11977 nnsf 13374 peano4nninf 13375 nninfalllemn 13377 nninfsellemeq 13385 |
Copyright terms: Public domain | W3C validator |