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Mirrors > Home > ILE Home > Th. List > 2onn | GIF version |
Description: The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) |
Ref | Expression |
---|---|
2onn | ⊢ 2𝑜 ∈ ω |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2o 6182 | . 2 ⊢ 2𝑜 = suc 1𝑜 | |
2 | 1onn 6279 | . . 3 ⊢ 1𝑜 ∈ ω | |
3 | peano2 4410 | . . 3 ⊢ (1𝑜 ∈ ω → suc 1𝑜 ∈ ω) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ suc 1𝑜 ∈ ω |
5 | 1, 4 | eqeltri 2160 | 1 ⊢ 2𝑜 ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 suc csuc 4192 ωcom 4405 1𝑜c1o 6174 2𝑜c2o 6175 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-uni 3654 df-int 3689 df-suc 4198 df-iom 4406 df-1o 6181 df-2o 6182 |
This theorem is referenced by: 3onn 6281 nn2m 6285 isomnimap 6793 enomnilem 6794 fodjuomnilemf 6800 infnninf 6805 nnnninf 6806 exmidfodomrlemr 6828 exmidfodomrlemrALT 6829 prarloclemarch2 6978 nq02m 7024 prarloclemlt 7052 prarloclemlo 7053 prarloclem3 7056 prarloclemn 7058 prarloclem5 7059 prarloclemcalc 7061 hash3 10221 0nninf 11893 nnsf 11895 nninfex 11901 nninfsellemdc 11902 nninfself 11905 |
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