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Theorem onsuci 4370
Description: The successor of an ordinal number is an ordinal number. Corollary 7N(c) of [Enderton] p. 193. (Contributed by NM, 12-Jun-1994.)
Hypothesis
Ref Expression
onssi.1 𝐴 ∈ On
Assertion
Ref Expression
onsuci suc 𝐴 ∈ On

Proof of Theorem onsuci
StepHypRef Expression
1 onssi.1 . 2 𝐴 ∈ On
2 suceloni 4355 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
31, 2ax-mp 7 1 suc 𝐴 ∈ On
Colors of variables: wff set class
Syntax hints:  wcel 1448  Oncon0 4223  suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-on 4228  df-suc 4231
This theorem is referenced by:  ordtri2orexmid  4376  onsucsssucexmid  4380  ordsoexmid  4415  ordtri2or2exmid  4424  tfr0dm  6149  1on  6250  2on  6252  3on  6254  4on  6255  prarloclemarch2  7128
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