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Theorem pion 10023
Description: A positive integer is an ordinal number. (Contributed by NM, 23-Mar-1996.) (New usage is discouraged.)
Assertion
Ref Expression
pion (𝐴N𝐴 ∈ On)

Proof of Theorem pion
StepHypRef Expression
1 pinn 10022 . 2 (𝐴N𝐴 ∈ ω)
2 nnon 7337 . 2 (𝐴 ∈ ω → 𝐴 ∈ On)
31, 2syl 17 1 (𝐴N𝐴 ∈ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  Oncon0 5967  ωcom 7331  Ncnpi 9988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-tr 4978  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-om 7332  df-ni 10016
This theorem is referenced by:  indpi  10051  nqereu  10073
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