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Theorem 3onn 6562
Description: The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
3onn 3o ∈ ω

Proof of Theorem 3onn
StepHypRef Expression
1 df-3o 6458 . 2 3o = suc 2o
2 2onn 6561 . . 3 2o ∈ ω
3 peano2 4619 . . 3 (2o ∈ ω → suc 2o ∈ ω)
42, 3ax-mp 5 . 2 suc 2o ∈ ω
51, 4eqeltri 2262 1 3o ∈ ω
Colors of variables: wff set class
Syntax hints:  wcel 2160  suc csuc 4390  ωcom 4614  2oc2o 6450  3oc3o 6451
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 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458
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 2758  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-uni 3832  df-int 3867  df-suc 4396  df-iom 4615  df-1o 6456  df-2o 6457  df-3o 6458
This theorem is referenced by:  4onn  6563  hash4  10859
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