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Theorem 4onn 6349
Description: The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
4onn 4o ∈ ω

Proof of Theorem 4onn
StepHypRef Expression
1 df-4o 6246 . 2 4o = suc 3o
2 3onn 6348 . . 3 3o ∈ ω
3 peano2 4447 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 7 . 2 suc 3o ∈ ω
51, 4eqeltri 2172 1 4o ∈ ω
Colors of variables: wff set class
Syntax hints:  wcel 1448  suc csuc 4225  ωcom 4442  3oc3o 6238  4oc4o 6239
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-iom 4443  df-1o 6243  df-2o 6244  df-3o 6245  df-4o 6246
This theorem is referenced by: (None)
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