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Theorem 4onn 8684
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 8510 . 2 4o = suc 3o
2 3onn 8683 . . 3 3o ∈ ω
3 peano2 7913 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 5 . 2 suc 3o ∈ ω
51, 4eqeltri 2836 1 4o ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  suc csuc 6385  ωcom 7888  3oc3o 8502  4oc4o 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-om 7889  df-1o 8507  df-2o 8508  df-3o 8509  df-4o 8510
This theorem is referenced by:  4finon  43470
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