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Theorem 4onn 8682
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 8508 . 2 4o = suc 3o
2 3onn 8681 . . 3 3o ∈ ω
3 peano2 7913 . . 3 (3o ∈ ω → suc 3o ∈ ω)
42, 3ax-mp 5 . 2 suc 3o ∈ ω
51, 4eqeltri 2835 1 4o ∈ ω
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  suc csuc 6388  ωcom 7887  3oc3o 8500  4oc4o 8501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-om 7888  df-1o 8505  df-2o 8506  df-3o 8507  df-4o 8508
This theorem is referenced by:  4finon  43442
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