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Theorem suceloni 4420
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 4300 . . 3 (𝐴 ∈ On → Ord 𝐴)
2 ordsucim 4419 . . 3 (Ord 𝐴 → Ord suc 𝐴)
31, 2syl 14 . 2 (𝐴 ∈ On → Ord suc 𝐴)
4 sucexg 4417 . . 3 (𝐴 ∈ On → suc 𝐴 ∈ V)
5 elong 4298 . . 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 1480  Vcvv 2686  Ord word 4287  Oncon0 4288  suc csuc 4290
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3740  df-tr 4030  df-iord 4291  df-on 4293  df-suc 4296
This theorem is referenced by:  sucelon  4422  unon  4430  onsuci  4435  ordsucunielexmid  4449  tfrlemisucaccv  6225  tfrexlem  6234  tfri1dALT  6251  rdgisuc1  6284  rdgon  6286  oacl  6359  oasuc  6363  omsuc  6371
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