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Theorem 4onn 8266
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 8103 . 2 4o = suc 3o
2 3onn 8265 . . 3 3o ∈ ω
3 peano2 7598 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 5 . 2 suc 3o ∈ ω
51, 4eqeltri 2912 1 4o ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  suc csuc 6182  ωcom 7576  3oc3o 8095  4oc4o 8096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-tr 5160  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-om 7577  df-1o 8100  df-2o 8101  df-3o 8102  df-4o 8103
This theorem is referenced by: (None)
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