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Theorem 4onn 8006
 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 7846 . 2 4o = suc 3o
2 3onn 8005 . . 3 3o ∈ ω
3 peano2 7364 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 5 . 2 suc 3o ∈ ω
51, 4eqeltri 2854 1 4o ∈ ω
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2106  suc csuc 5978  ωcom 7343  3oc3o 7838  4oc4o 7839 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-om 7344  df-1o 7843  df-2o 7844  df-3o 7845  df-4o 7846 This theorem is referenced by: (None)
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