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Theorem 4onn 7707
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 7548 . 2 4𝑜 = suc 3𝑜
2 3onn 7706 . . 3 3𝑜 ∈ ω
3 peano2 7071 . . 3 (3𝑜 ∈ ω → suc 3𝑜 ∈ ω)
42, 3ax-mp 5 . 2 suc 3𝑜 ∈ ω
51, 4eqeltri 2695 1 4𝑜 ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 1988  suc csuc 5713  ωcom 7050  3𝑜c3o 7540  4𝑜c4o 7541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-om 7051  df-1o 7545  df-2o 7546  df-3o 7547  df-4o 7548
This theorem is referenced by: (None)
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