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Theorem suceloni 4387
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 4267 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsucim 4386 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 sucexg 4384 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
5 elong 4265 . . 3 (suc 𝐴 ∈ V → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
64, 5syl 14 . 2 (𝐴 ∈ On → (suc 𝐴 ∈ On ↔ Ord suc 𝐴))
73, 6mpbird 166 1 (𝐴 ∈ On → suc 𝐴 ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1465  Vcvv 2660  Ord word 4254  Oncon0 4255  suc csuc 4257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263
This theorem is referenced by:  sucelon  4389  unon  4397  onsuci  4402  ordsucunielexmid  4416  tfrlemisucaccv  6190  tfrexlem  6199  tfri1dALT  6216  rdgisuc1  6249  rdgon  6251  oacl  6324  oasuc  6328  omsuc  6336
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