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Theorem suceloni 4255
Description: The successor of an ordinal number is an ordinal number. Proposition 7.24 of [TakeutiZaring] p. 41. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
suceloni (𝐴 ∈ On → suc 𝐴 ∈ On)

Proof of Theorem suceloni
StepHypRef Expression
1 eloni 4140 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsucim 4254 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 sucexg 4252 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
5 elong 4138 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
64, 5syl 14 . 2 (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
73, 6mpbird 160 1 (𝐴 ∈ On → suc 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  Vcvv 2574  Ord word 4127  Oncon0 4128  suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136
This theorem is referenced by:  sucelon  4257  unon  4265  onsuci  4270  ordsucunielexmid  4284  tfrlemisucaccv  5970  tfrexlem  5979  rdgisuc1  6002  frecsuclemdm  6019  oacl  6071  oasuc  6075  omsuc  6082
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