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Theorem 4onn 6426
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 6323 . 2 4o = suc 3o
2 3onn 6425 . . 3 3o ∈ ω
3 peano2 4516 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 5 . 2 suc 3o ∈ ω
51, 4eqeltri 2213 1 4o ∈ ω
Colors of variables: wff set class
Syntax hints:  wcel 1481  suc csuc 4294  ωcom 4511  3oc3o 6315  4oc4o 6316
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 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
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 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-int 3779  df-suc 4300  df-iom 4512  df-1o 6320  df-2o 6321  df-3o 6322  df-4o 6323
This theorem is referenced by: (None)
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