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Theorem onsuci 4308
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 4293 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
31, 2ax-mp 7 1 suc 𝐴 ∈ On
Colors of variables: wff set class
Syntax hints:  wcel 1436  Oncon0 4166  suc csuc 4168
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-uni 3639  df-tr 3914  df-iord 4169  df-on 4171  df-suc 4174
This theorem is referenced by:  ordtri2orexmid  4314  onsucsssucexmid  4318  ordsoexmid  4353  ordtri2or2exmid  4362  tfr0dm  6043  1on  6144  2on  6146  3on  6148  4on  6149  prarloclemarch2  6925
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