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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 6220 | . 2 ⊢ 2o = suc 1o | |
2 | 1onn 6319 | . . 3 ⊢ 1o ∈ ω | |
3 | peano2 4438 | . . 3 ⊢ (1o ∈ ω → suc 1o ∈ ω) | |
4 | 2, 3 | ax-mp 7 | . 2 ⊢ suc 1o ∈ ω |
5 | 1, 4 | eqeltri 2167 | 1 ⊢ 2o ∈ ω |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1445 suc csuc 4216 ωcom 4433 1oc1o 6212 2oc2o 6213 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-uni 3676 df-int 3711 df-suc 4222 df-iom 4434 df-1o 6219 df-2o 6220 |
This theorem is referenced by: 3onn 6321 nn2m 6325 isomnimap 6880 enomnilem 6881 fodjuf 6888 infnninf 6893 nnnninf 6894 ismkvmap 6898 exmidfodomrlemr 6925 exmidfodomrlemrALT 6926 prarloclemarch2 7075 nq02m 7121 prarloclemlt 7149 prarloclemlo 7150 prarloclem3 7153 prarloclemn 7155 prarloclem5 7156 prarloclemcalc 7158 hash3 10336 0nninf 12598 nnsf 12600 nninfex 12606 nninfsellemdc 12607 nninfself 12610 |
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